# A classification of the Wadge hierarchies
on zero-dimensional Polish spaces

Raphaël Carroy  Dipartimento di matematica «Giuseppe Peano», Università di
Torino, Via Carlo Alberto 10, 10121 Torino — Italy<EMAIL_ADDRESS>,
Luca Motto Ros  Dipartimento di matematica «Giuseppe Peano», Università di
Torino, Via Carlo Alberto 10, 10121 Torino — Italy<EMAIL_ADDRESS>and
Salvatore Scamperti  Dipartimento di matematica «Giuseppe Peano», Università
di Torino, Via Carlo Alberto 10, 10121 Torino — Italy
<EMAIL_ADDRESS>
###### Abstract.

We provide a complete classification, up to order-isomorphism, of all possible
Wadge hierarchies on zero-dimensional Polish spaces using (essentially)
countable ordinals as complete invariants. We also observe that although our
assignment of invariants is very simple and there are only $\aleph_{1}$-many
equivalence classes, the above classification problem is quite complex from
the descriptive set-theoretic point of view: in particular, there is no Borel
procedure to determine whether two zero-dimensional Polish spaces have
isomorphic Wadge hierarchies. All results are based on a complete and explicit
description of the Wadge hierarchy on an arbitrary zero-dimensional Polish
space, depending on its topological properties.

###### Key words and phrases:

Wadge reducibility, continuous reducibility, Wadge quasi-order, zero-
dimensional Polish spaces

###### 2020 Mathematics Subject Classification:

03E15, 54E50

This work was supported by the project PRIN 2017 “Mathematical Logic: models,
sets, computability” (prot. 2017NWTM8R) and by the “National Group for
Algebraic and Geometric Structures, and their Applications” (GNSAGA – INDAM)

The authors are grateful to the anonymous referee for their useful suggestions
and references

##  1\. Introduction

Work in ${\sf ZF}+{\sf DC}(\mathbb{R})$.111${\sf DC}(\mathbb{R})$ is the axiom
of Dependent Choices over $\mathbb{R}$, which is equivalent to having ${\sf
DC}(X)$ for all Polish spaces $X$. Given topological spaces $X$ and $Y$, a
_continuous reduction_ from $A\subseteq X$ to $B\subseteq Y$ is a continuous
function $f\colon X\to Y$ satisfying $f^{-1}(B)=A$. When such an $f$ exists we
say that $A\subseteq X$ _continuously reduces_ or _Wadge reduces_ to
$B\subseteq Y$, and we write $A\leq_{\mathsf{W}}^{X,Y}B$, or
$A\leq_{\mathsf{W}}^{X}B$ if $X=Y$. Notice that $A\leq_{\mathsf{W}}^{X,Y}B$ if
and only if ${X\setminus A}\leq_{\mathsf{W}}^{X,Y}{X\setminus B}$. We also
denote by $<_{\mathsf{W}}^{X,Y}$ the strict part of $\leq_{\mathsf{W}}^{X,Y}$,
that is:
$A<_{\mathsf{W}}^{X,Y}B\iff{{A\leq_{\mathsf{W}}^{X,Y}B}\wedge{B\not\leq_{\mathsf{W}}^{Y,X}A}}$.
Similarly, we write
$A\equiv_{\mathsf{W}}^{X,Y}B\iff{{A\leq_{\mathsf{W}}^{X,Y}B}\wedge{B\leq_{\mathsf{W}}^{Y,X}A}}$.
To simplify the notation, we again write $<_{\mathsf{W}}^{X}$ and
$\equiv_{\mathsf{W}}^{X}$ instead of $<_{\mathsf{W}}^{X,X}$ and
$\equiv_{\mathsf{W}}^{X,X}$, respectively.

Continuous reducibility is a transitive and reflexive relation, that is a
_quasi-order_ , and $\equiv_{\mathsf{W}}^{X}$ is the equivalence relation
canonically induced by $\leq_{\mathsf{W}}^{X}$. We call _Wadge degree_ of
$A\subseteq X$ the $\equiv_{\mathsf{W}}^{X}$-equivalence class of $A$, and we
denote it by $[A]_{\mathsf{W}}^{X}$. The Wadge reducibility
$\leq_{\mathsf{W}}^{X}$ induces a partial order on Wadge degrees of $X$ that
we call _Wadge hierarchy_ on $X$ and denote by $\mathscr{W}_{X}$. When
$\leq_{\mathsf{W}}^{X}$ is well-founded, we can associate to each set
$A\subseteq X$ (or to its Wadge degree $[A]_{\mathsf{W}}^{X}$) its rank
according to $\leq_{\mathsf{W}}^{X}$, which is called the _Wadge rank_ of $A$
(or of $[A]_{\mathsf{W}}^{X}$) and is denoted by $||A||_{\mathsf{W}}^{X}$. For
technical and historical reasons, we use $1$ as the minimal value for
$||A||_{\mathsf{W}}^{X}$. We also let $\Theta_{X}$ be the length of
$\mathscr{W}_{X}$, i.e. $\Theta_{X}=\sup\\{||A||_{\mathsf{W}}^{X}+1\mid
A\subseteq X\\}$.

To simplify the notation, given a set $A\subseteq X$ we sometimes write $\neg
A$ instead of $X\setminus A$ when the ambient space $X$ is clear from the
context. Wadge left his name to continuous reducibility when he proved in
[Wad83], by game theoretic methods, that $\leq_{\mathsf{W}}^{\mathcal{N}}$ on
Borel subsets of the Baire space $\mathcal{N}={}^{\omega}\omega$ is well-
founded and satisfies the _Semi-Linear Ordering principle_ , or
$\mathsf{SLO}^{\mathsf{W}}_{\mathcal{N}}(\mathrm{Bor})$: for all (Borel) sets
$A,B\subseteq\mathcal{N}$

(1.1) $A\leq_{\mathsf{W}}^{\mathcal{N}}B\quad\text{or}\quad\neg
B\leq_{\mathsf{W}}^{\mathcal{N}}A.$

These two results generalize under the _Axiom of Determinacy_ (${\sf AD}$),
yielding to the fact that the Wadge hierarchy is _semi-well-ordered_ , namely:
the full $\mathsf{SLO}^{\mathsf{W}}_{\mathcal{N}}$ holds (i.e. (1.1) is true
for arbitrary subsets $A,B\subseteq\mathcal{N}$), and
$\leq_{\mathsf{W}}^{\mathcal{N}}$ is well-founded. (The former directly
follows from Wadge’s original lemma, while the latter is due to a highly
nontrivial argument by Martin and Monk.)

Early investigations on Wadge theory [Wad83, VW77, Ste77, Sol78] have fully
described the Wadge hierarchy on $\mathcal{N}$ under ${\sf AD}$ (see Figure
1). Indeed, by $\mathsf{SLO}^{\mathsf{W}}_{\mathcal{N}}$ one easily gets that
for each $A\subseteq\mathcal{N}$, either $A\leq_{\mathsf{W}}^{\mathcal{N}}\neg
A$, in which case we say that $A$ (or its Wadge degree
$[A]_{\mathsf{W}}^{\mathcal{N}}$) is _selfdual_ , or else $A$ and $\neg A$ are
$\leq_{\mathsf{W}}^{\mathcal{N}}$-incomparable, in which case $A$ is called
_nonselfdual_ and $\\{[A]_{\mathsf{W}}^{\mathcal{N}},[\neg
A]_{\mathsf{W}}^{\mathcal{N}}\\}$ is a maximal antichain, called a
_nonselfdual pair_. The minimal Wadge degrees are those in the nonselfdual
pair
$\\{[\mathcal{N}]_{\mathsf{W}}^{\mathcal{N}},[\emptyset]_{\mathsf{W}}^{\mathcal{N}}\\}$,
and indeed $[\mathcal{N}]_{\mathsf{W}}^{\mathcal{N}}=\\{\mathcal{N}\\}$ and
$[\emptyset]_{\mathsf{W}}^{\mathcal{N}}=\\{\emptyset\\}$. After this minimal
nonselfdual pair there is a single selfdual degree consisting of all
nontrivial clopen subsets of $\mathcal{N}$. Then nonselfdual pairs and
selfdual degrees alternate, at limit levels of countable cofinality there is
always a selfdual degree, while at limit levels of uncountable cofinality
there is always a nonselfdual pair. Finally, $\Theta_{\mathcal{N}}=\Theta$
where

$\Theta=\sup\\{\alpha\in\mathrm{Ord}\mid\mathbb{R}\text{ surjects onto
}\alpha\\}.$

$\dotsc$$\dotsc$$\dotsc$$\mathrm{cof}=\omega$$\mathrm{cof}>\omega$$\Theta_{\mathcal{N}}=\Theta$
Figure 1. The Wadge hierarchy $\mathscr{W}_{\mathcal{N}}$ on the Baire space
$\mathcal{N}$.

Wadge theory has received a lot of attention, as witnessed by [VW78, Lou83,
LSR88, Dup01, AM03, And07, MR09, MR10b, MR10a, KM19, CMM20], among many other
works. Several generalizations have been considered, including variations of
the reducibility (usually replacing continuous functions with other natural
classes of functions) and/or replacing $\mathcal{N}$ with other spaces. In the
second direction, the first natural move is to consider other zero-dimensional
Polish spaces.222Recall that a space is _Polish_ if it is separable and
completely metrizable, and _zero-dimensional_ if it has a basis of clopen
sets. For example, if one assumes ${\sf AD}$ again and considers the Cantor
space $\mathcal{C}={}^{\omega}2$, then the corresponding Wadge hierarchy
$\mathscr{W}_{\mathcal{C}}$ looks exactly like $\mathscr{W}_{\mathcal{N}}$
except for the fact that at limit levels one always has a nonselfdual pair,
independently of the cofinality of the level (see Figure 2). This is obtained
via straightforward modifications of the results and game-theoretic methods
used in the case of $\mathcal{N}$.333Although this was probably known in the
field since the Eighties, to the best of our knowledge the first complete
accounts on the structure $\mathscr{W}_{\mathcal{C}}$ appeared only more
recently in [And07, Section 2.7] and [AC13, Section 4].

$\dotsc$$\dotsc$$\dotsc$$\mathrm{cof}=\omega$$\mathrm{cof}>\omega$$\Theta_{\mathcal{C}}=\Theta$
Figure 2. The Wadge hierarchy $\mathscr{W}_{\mathcal{C}}$ on the Cantor space
$\mathcal{C}$.

If one moves to an arbitrary zero-dimensional Polish space $X$ (equivalently:
to an arbitrary closed subspace of $\mathcal{N}$) different from $\mathcal{N}$
and $\mathcal{C}$, instead, very little is known about how $\mathscr{W}_{X}$
looks like. The hierarchy starts again with the nonselfdual pair
$\\{[X]_{\mathsf{W}}^{X},[\emptyset]_{\mathsf{W}}^{X}\\}$, where
$[X]_{\mathsf{W}}^{X}=\\{X\\}$ and
$[\emptyset]_{\mathsf{W}}^{X}=\\{\emptyset\\}$, followed by the selfdual
degree of all nontrivial clopen subsets of $X$. As discussed in Section 2.2,
the existence of a retraction from $\mathcal{N}$ onto $X$ yields the fact
that, still under ${\sf AD}$, the Wadge hierarchy $\mathscr{W}_{X}$ is semi-
well-ordered, i.e. the quasi-order $\leq_{\mathsf{W}}^{X}$ is well-founded and
$\mathsf{SLO}^{\mathsf{W}}_{X}$ is true, i.e.
${A\leq_{\mathsf{W}}^{X}B}\vee{\neg B\leq_{\mathsf{W}}^{X}A}$ holds for all
$A,B\subseteq X$ (see also [MRSS15]). This implies that at each level of
$\mathscr{W}_{X}$ we find either a nonselfdual pair
$\\{[A]_{\mathsf{W}}^{X},[\neg A]_{\mathsf{W}}^{X}\\}$ (when
$A\not\leq_{\mathsf{W}}^{X}\neg A$) or a single selfdual degree
$[A]_{\mathsf{W}}^{X}$ (when $A\leq_{\mathsf{W}}^{X}\neg A$). Another easy
observation is that $\Theta_{X}=\Theta_{\mathcal{N}}=\Theta$ if $X$ is
uncountable, while $\Theta_{X}<\omega_{1}$ if $X$ is countable. This follows
from the retraction trick again (see Section 2.2) if $X$ is uncountable,
because in this case $X$ contains a closed homeomorphic copy of $\mathcal{C}$;
if instead $X$ is countable, we use the fact that every subset of $X$ is a
$\boldsymbol{\Delta}^{0}_{2}$-set, and indeed it is in
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1})$, where $1\leq\alpha<\omega_{1}$
depends on the Cantor-Bendixson rank $||X||_{\operatorname{CB}}$ of $X$.
However, nothing else is known, and a complete and detailed description of
$\mathscr{W}_{X}$ has never been carried out in the literature: the main
purpose of this article is to fill this gap and prove the following result.
(Notice that in the countable case zero-dimensionality is automatic and could
thus be removed from the hypotheses, and moreover in such case we do not need
any determinacy assumptions because
$\mathscr{P}(X)\subseteq\boldsymbol{\Delta}^{0}_{2}(X)$ and thus Martin’s
Borel determinacy is enough.)

######  Main Theorem 1.

Let $X$ be an arbitrary zero-dimensional Polish space, and assume ${\sf AD}$
if $X$ is uncountable. Then

  1. (1)

$\mathscr{W}_{X}$ is semi-well-ordered (Proposition 2.3);

  2. (2)

selfdual degrees and nonselfdual pairs alternate, starting with the
nonselfdual pair $\\{[X]_{\mathsf{W}}^{X},[\emptyset]_{\mathsf{W}}^{X}\\}$ at
the bottom, followed by the selfdual degree of all nontrivial clopen sets
(Theorem 4.4, Theorem 4.6, and Proposition 2.2);

  3. (3)

for limit levels, we distinguish two cases:

    1. (a)

if the perfect kernel of $X$ is not compact (hence nonempty), then at all
limit levels of uncountable cofinality there is a nonselfdual pair and at all
limit levels of countable cofinality there is a selfdual degree (Proposition
4.7 and Theorem 4.9);

    2. (b)

otherwise there is $\alpha_{X}<\omega_{1}$ such that for all limit
$\alpha<\Theta_{X}$, there is a selfdual degree at level $\alpha$ if and only
if $\alpha<\alpha_{X}$; in particular, $\alpha_{X}=0$ if $X$ is compact
(Corollary 4.14 and the ensuing discussion);

  4. (4)

$\Theta_{X}=\Theta$ if $X$ is uncountable, while if $X$ is countable then
$\Theta_{X}=2\cdot||X||_{\operatorname{CB}}+\varepsilon_{X}$ for some
$\varepsilon_{X}\in\\{-1,0,1\\}$ (Proposition 2.5 and Theorem 4.15).

The parameter $\alpha_{X}$ depends on a new rank on $X$ which we call _compact
rank_ and denote by $||X||_{\operatorname{Comp}}$ (Section 3.3). As for item
(4), the ordinal $||X||_{\operatorname{CB}}$ is the usual Cantor-Bendixson
rank of $X$, while the parameter $\varepsilon_{X}$ depends on
$||X||_{\operatorname{CB}}$, $\alpha_{X}$, and whether $X$ is simple or not
(see Section 2.5 for the relevant definitions).

It follows from 1 that if $X$ has a non-compact perfect kernel, then
$\mathscr{W}_{X}$ is isomorphic to $\mathscr{W}_{\mathcal{N}}$ (Figure 1); if
$X$ is compact and, more generally, if $||X||_{\operatorname{Comp}}<\omega$,
then $\mathscr{W}_{X}$ is isomorphic to $\mathscr{W}_{\mathcal{C}}$ (Figure
2); in the remaining cases, i.e. when $\mathscr{W}_{X}$ is isomorphic neither
to $\mathscr{W}_{\mathcal{N}}$ nor to $\mathscr{W}_{\mathcal{C}}$, the
hierarchy $\mathscr{W}_{X}$ is a mixture of the two, with an initial segment
(of countable length) which is Baire-like and the rest of the hierarchy which
is Cantor-like (see Figure 3).

$\dotsc$$\dotsc$$\dotsc$$\dotsc$any limitany
limit$\alpha_{X}$$\Theta_{X}\in\omega_{1}\cup\\{\Theta\\}$Baire-likeCantor-
like Figure 3. The Wadge hierarchy $\mathscr{W}_{X}$ on an arbitrary zero-
dimensional Polish space $X$ when $\mathscr{W}_{X}$ is isomorphic neither to
$\mathscr{W}_{\mathcal{N}}$ nor to $\mathscr{W}_{\mathcal{C}}$. In this case,
the ordinal $\alpha_{X}$ satisfies $\omega<\alpha_{X}<\omega_{1}$.

1 is optimal, in the sense that for all possibilities which are coherent with
the limitations contained therein, there is a zero-dimensional Polish space
$X$ whose Wadge hierarchy $\mathscr{W}_{X}$ realizes precisely that scenario
(Section 4.4). Thus what we have obtained is a complete classification (up to
order-isomorphism) of all possible Wadge hierarchies for zero-dimensional
Polish spaces, where the complete invariants are given by pairs of ordinals
$(\alpha_{X},\Theta_{X})$ which can easily be coded as countable ordinals (see
the discussion at the beginning of Section 5).

Although our classification of the possible Wadge hierarchies over zero-
dimensional Polish spaces $X$ is complete and quite satisfactory, we also show
that it is not possible to determine $\mathscr{W}_{X}$ in a Borel manner. For
example, by Theorem 5.3 the collection of all zero-dimensional Polish spaces
whose Wadge hierarchy is order-isomorphic to $\mathscr{W}_{\mathcal{N}}$ is a
complete analytic subset of the Effros Borel space $F(\mathcal{N})$. More
generally, for all nontrivial zero-dimensional Polish spaces $X$ there is no
Borel procedure to determine which zero-dimensional Polish spaces $Y$ give
rise to a Wadge hierarchy $\mathscr{W}_{Y}$ order-isomorphic to
$\mathscr{W}_{X}$.

######  Main Theorem 2.

Let $X$ be a zero-dimensional Polish space with at least two points, and
assume ${\sf AD}$ if $X$ is uncountable. Then the set of zero-dimensional
Polish spaces $Y$ such that $\mathscr{W}_{Y}$ is order-isomorphic to
$\mathscr{W}_{X}$ is not Borel in the Effros Borel space $F(\mathcal{N})$.

Indeed, it is not even possible to decide in a Borel way whether at a given
countable limit level we will find a nonselfdual pair or a selfdual degree, or
to determine the value of $\alpha_{X}$ and/or $\Theta_{X}$ (see Remark 5.4).
The reason behind this wild behaviour actually relies on the possible
configurations of isolated points. If we restrict to perfect spaces, then the
situation is radically neater and we have only two (Borel) possibilities:
spaces with a Baire-like Wadge hierarchy, and spaces with a Cantor-like Wadge
hierarchy (Proposition 5.5).

Finally, let us briefly discuss the optimality of the various hypotheses in
our results. As it is well-known in the area, the determinacy assumption ${\sf
AD}$ can be relaxed to get level-by-level statements, at the cost of making
such statements and the various definitions a lot more cumbersome. For a
detailed discussion on this matter, and specifically on nice pointclasses, see
[CMM22, Section 3]. In particular, if we restrict the attention to the Borel
subsets of $X$, then Martin’s Borel determinacy suffices and we do not need to
assume ${\sf AD}$ anywhere.

Moreover, we point out that there is a reason for restricting to zero-
dimensional Polish spaces. The Wadge quasi-order has been studied in various
other contexts, even beyond the realm of Polish spaces (see for instance
[Sel05, MRSS15, Peq15, IST19, Cam19, CM21]), but in most cases it is not semi-
well-ordered if the space is not zero-dimensional (with a few exceptions: see
[CM21, Theorem 33]). If one restricts the attention to Polish spaces, this
becomes even a characterization because $\mathscr{W}_{X}$ has infinite
antichains whenever $X$ is a metric (not necessarily complete or separable)
space with positive dimension (see [Sch18, Theorem 1.5]). Moreover, [Her93]
and [DV20] show among other things that $\mathscr{W}_{X}$ is ill-founded when
$X$ is $\mathbb{R}$ or the Scott domain, respectively. See [MRSS15] for more
possible behaviours and a rough classification of Wadge-like hierarchies on
arbitrary (quasi-)Polish spaces.

##  2\. Preliminaries and known facts

The monograph [Kec95] is our reference for all classical results and notation
in descriptive set theory. From this point onward, unless otherwise stated, we
assume ${\sf AD}$ and that $X$, $Y$, $Z$, …are nonempty zero-dimensional
Polish spaces. Recall however that the determinacy assumption is just for ease
of exposition, and that no further assumption is required if all results and
definitions are restricted to the realm of Borel sets.

###  2.1. Some easy and well-known facts

The principle $\mathsf{SLO}^{\mathsf{W}}_{X}$ can be reformulated to deal with
sets in possibly different zero-dimensional Polish spaces ([Kec95, Theorem
21.14]). The proof is identical to the original one by Wadge, except that the
rules of the so-called Wadge’s game are modified so that the players stay
inside their respective closed subspaces of $\mathcal{N}$.

######  Proposition 2.1 (${\sf AD}$).

Let $X,Y$ be zero-dimensional Polish spaces. Then for all $A\subseteq X$ and
$B\subseteq Y$

$A\leq_{\mathsf{W}}^{X,Y}B\quad\vee\quad\neg B\leq_{\mathsf{W}}^{Y,X}A.$

Notice that when $X=Y$, Proposition 2.1 is simply the assertion that
$\mathsf{SLO}^{\mathsf{W}}_{X}$ holds under ${\sf AD}$ for an arbitrary zero-
dimensional Polish space $X$. Proposition 2.1 is so crucial in the theory that
we will simply refer to it as _Wadge’s Lemma_.

Let $\mathrm{Set}_{\operatorname{SD}}(X)$ be the collection of all _selfdual
sets_ of $X$, i.e.

$\mathrm{Set}_{\operatorname{SD}}(X)=\\{A\subseteq X\mid
A\leq_{\mathsf{W}}^{X}\neg A\\},$

and, dually, let $\mathrm{Set}_{\operatorname{NSD}}(X)$ be the collection of
_nonselfdual sets_ of $X$, that is

$\mathrm{Set}_{\operatorname{NSD}}(X)=\mathscr{P}(X)\setminus\mathrm{Set}_{\operatorname{SD}}(X)=\\{A\subseteq
X\mid A\not\leq_{\mathsf{W}}^{X}\neg A\\}.$

Then Wadge’s Lemma implies that at each level of $\mathscr{W}_{X}$ we find
either a $2$-sized maximal antichain $\\{[A]_{\mathsf{W}}^{X},[\neg
A]_{\mathsf{W}}^{X}\\}$ for some $A\in\mathrm{Set}_{\operatorname{NSD}}(A)$,
called a _nonselfdual pair_ , or a single selfdual degree
$[A]_{\mathsf{W}}^{X}$ for some $A\in\mathrm{Set}_{\operatorname{SD}}(X)$.

Another easy observation (see [Cam19, Lemma 1]) is that for every nonempty
topological space $X$ the pair
$\\{[X]_{\mathsf{W}}^{X},[\emptyset]_{\mathsf{W}}^{X}\\}$ is nonselfdual and
its elements are minimal in $\mathscr{W}_{X}$; if $X$ is not connected then
right after such pair there is a single selfdual degree. (The proof of the
next proposition is left to the reader.)

######  Proposition 2.2.

Let $X$ be an arbitrary nonempty topological space. Then
$[X]_{\mathsf{W}}^{X}=\\{X\\}$ and
$[\emptyset]_{\mathsf{W}}^{X}=\\{\emptyset\\}$. Moreover, the pair
$\\{[X]_{\mathsf{W}}^{X},[\emptyset]_{\mathsf{W}}^{X}\\}$ is nonselfdual and
$\leq_{\mathsf{W}}^{X}$-minimal in $\mathscr{W}_{X}$, that is, for every
$\emptyset,X\neq A\subseteq X$ we have $\emptyset,X<_{\mathsf{W}}^{X}A$.

If moreover $X$ is disconnected, then immediately above
$\\{[X]_{\mathsf{W}}^{X},[\emptyset]_{\mathsf{W}}^{X}\\}$ there is a single
selfdual degree consisting of all nontrivial clopen subsets of $X$, and all
other Wadge degrees are strictly $\leq_{\mathsf{W}}^{X}$-above it.

###  2.2. The retraction method

Part (1) of 1 can be proved using a trick due to Marcone (unpublished, but see
[And07, Proposition 28]) and, independently, to Selivanov [Sel05, Corollary
2.4]. If $X\subseteq Y$ are such that there is a retraction444A retraction
from $Y$ onto $X$ is a (surjective) continuous function $\rho\colon Y\to X$
such that $\rho\restriction X$ is the identity on $X$. $\rho$ from $Y$ onto
$X$, then the map from $\mathscr{P}(X)$ to $\mathscr{P}(Y)$ defined by
$A\mapsto\rho^{-1}(A)$ embeds the structure
$\langle\mathscr{P}(X),{\leq_{\mathsf{W}}^{X}},{\neg}\rangle$ into
$\langle\mathscr{P}(Y),{\leq_{\mathsf{W}}^{Y}},{\neg}\rangle$ (see also
[MRSS15, Proposition 5.4] for a more general result). Notice that up to Wadge
equivalence, the choice of $\rho$ is irrelevant: if $\rho_{1},\rho_{2}\colon
Y\to X$ are two retractions then
$\rho_{1}^{-1}(A)\equiv_{\mathsf{W}}^{Y}\rho_{2}^{-1}(A)$, as witnessed by the
maps $\rho_{1}$ and $\rho_{2}$ themselves. Notice also that for any retraction
$\rho\colon Y\to X$ and $A\subseteq X$, we have
$A\equiv_{\mathsf{W}}^{X,Y}\rho^{-1}(A)$ via the inclusion map $\iota\colon
X\to Y$ and the retraction $\rho\colon Y\to X$.

It is a classical result that every zero-dimensional Polish space can be
construed as a closed subspace of $\mathcal{N}$, thus there is a retraction of
$\mathcal{N}$ onto $X$. Therefore we get the desired result ([MRSS15, Theorem
5.12(1)]).

######  Proposition 2.3 (${\sf AD}$).

Let $X$ be a zero-dimensional Polish space. Then
$\mathsf{SLO}^{\mathsf{W}}_{X}$ holds and $\leq_{\mathsf{W}}^{X}$ is well-
founded, i.e. $\mathscr{W}_{X}$ is semi-well-ordered.

Proposition 2.3 allows us to split ordinals $1\leq\alpha<\Theta_{X}$ in two
groups: those for which at level $\alpha$ of $\mathscr{W}_{X}$ we find a
single selfdual degree and those for which we instead find a nonselfdual pair.
This is condensed in the following definition and notation.

######  Definition 2.4.

An ordinal $1\leq\alpha<\Theta_{X}$ is called _selfdual_ for $X$ if the
subsets of $X$ with Wadge rank $\alpha$ are selfdual, otherwise we say that
$\alpha$ is _nonselfdual_. We denote by $\mathrm{Ord}_{\mathrm{SD}}(X)$ the
set of all $1\leq\alpha<\Theta_{X}$ which are selfdual for $X$, while
$\mathrm{Ord}_{\mathrm{NSD}}(X)$ stands for the set of nonselfdual
$1\leq\alpha<\Theta_{X}$.

Notice that by Proposition 2.3, determining the structure of $\mathscr{W}_{X}$
amounts to determine the two sets of ordinals $\mathrm{Ord}_{\mathrm{SD}}(X)$
and $\mathrm{Ord}_{\mathrm{NSD}}(X)$, together with the exact value of
$\Theta_{X}$.

The same trick based on retractions can also be used to prove the first half
of part (4) of 1, i.e. to compute $\Theta_{X}$ when $X$ is uncountable.
Indeed, if a zero-dimensional Polish space $X$ is uncountable, then
$\mathcal{C}$ is (homeomorphic to) a closed subset of $X$: therefore there is
a retraction of $X$ onto $\mathcal{C}$, and thus
$\langle\mathscr{P}(\mathcal{C}),{\leq_{\mathsf{W}}^{\mathcal{C}}},{\neg}\rangle$
embeds into $\langle\mathscr{P}(X),{\leq_{\mathsf{W}}^{X}},{\neg}\rangle$.
Combined with the fact that, as observed,
$\langle\mathscr{P}(X),{\leq_{\mathsf{W}}^{X}},{\neg}\rangle$ embeds into
$\langle\mathscr{P}(\mathcal{N}),{\leq_{\mathsf{W}}^{\mathcal{N}}},{\neg}\rangle$,
this shows that $\Theta_{\mathcal{C}}\leq\Theta_{X}\leq\Theta_{\mathcal{N}}$.
Since under ${\sf AD}$ it holds
$\Theta_{\mathcal{C}}=\Theta_{\mathcal{N}}=\Theta$, we get:

######  Proposition 2.5 (${\sf AD}$).

Let $X$ be an uncountable zero-dimensional Polish space. Then
$\mathscr{W}_{X}$ has length $\Theta_{X}=\Theta$.

Finally, combining the retraction method with a well-known result in Wadge
theory due to Wadge himself [And06, Lemma 3], we obtain a technical fact that
provides a useful characterization of selfdual sets.

######  Proposition 2.6 (${\sf AD}$).

Let $X$ be a zero-dimensional Polish space. Then $A\subseteq X$ is selfdual if
and only if there are a clopen partition $(V_{n})_{n\in\omega}$ of $X$ and
sets $(A_{n})_{n\in\omega}$ such that $A_{n}<_{\mathsf{W}}^{X}A$ and
$A=\bigcup_{n\in\omega}(A_{n}\cap V_{n})$.

Moreover, we can further assume that each $A_{n}$ is nonselfdual, and
$||A||_{\mathsf{W}}^{X}=\sup_{n\in\omega}(||A_{n}||_{\mathsf{W}}^{X}+1)$.

###### Proof.

The backward direction is obvious, as by $\mathsf{SLO}^{\mathsf{W}}_{X}$ we
have $A_{n}\leq_{\mathsf{W}}^{X}\neg A$ for all $n\in\omega$: if
$(f_{n})_{n\in\omega}$ are continuous maps witnessing this, then
$f=\bigcup_{n\in\omega}(f_{n}\restriction V_{n})$ is a continuous map
witnessing $A\leq_{\mathsf{W}}^{X}\neg A$.

Conversely, let $X$ be a closed subspace of $\mathcal{N}$ and
$\rho\colon\mathcal{N}\to X$ be a retraction. If $A$ is selfdual in $X$, then
$A^{\prime}=\rho^{-1}(A)$ is selfdual in $\mathcal{N}$. Then the set of those
$s\in\omega^{<\omega}$ such that
$A^{\prime}\leq_{\mathsf{W}}^{\mathcal{N},\boldsymbol{N}_{s}}A^{\prime}\cap\boldsymbol{N}_{s}$
is a nonempty well-founded tree
$\boldsymbol{T}(A^{\prime})\subseteq\omega^{<\omega}$ ([And03, Lemma 22]). Let
$(V_{n})_{n\in\omega}$ be an enumeration of the sets of the form
$\boldsymbol{N}_{s{}^{\smallfrown}{}k}\cap X$ with $k\in\omega$ and $s$ a leaf
in $\boldsymbol{T}(A^{\prime})$, and set $A_{n}=X$ if $V_{n}\subseteq A$ and
$A_{n}=A\cap V_{n}$ otherwise. It is easy to check that the $V_{n}$’s and
$A_{n}$’s are as required.

The additional part on nonselfduality of the $A_{n}$’s follows by iterating
the construction on those $A_{n}$ which happen to be selfdual in $X$. (This
process must terminate on each branch after finitely many steps because
otherwise we would construct a strictly $\leq_{\mathsf{W}}^{X}$-decreasing
sequence of subsets of $X$, against Proposition 2.3.)

Finally, set $\alpha=||A||_{\mathsf{W}}^{X}$ and
$\alpha_{n}=||A_{n}||_{\mathsf{W}}^{X}$, so that in particular
$\alpha_{n}<\alpha$. If $B\subseteq X$ is a set with
$||B||_{\mathsf{W}}^{X}=\sup_{n\in\omega}(\alpha_{n}+1)$, then
$A_{n}<_{\mathsf{W}}^{X}B$ because
$||A_{n}||_{\mathsf{W}}^{X}=\alpha_{n}<\alpha_{n}+1\leq||B||_{\mathsf{W}}^{X}$:
if $f_{n}$ witnesses $A_{n}\leq_{\mathsf{W}}^{X}B$, then
$A\leq_{\mathsf{W}}^{X}B$ via $\bigcup_{n\in\omega}(f_{n}\restriction V_{n})$,
thus
$\sup_{n\in\omega}(\alpha_{n}+1)=||B||_{\mathsf{W}}^{X}\geq||A||_{\mathsf{W}}^{X}=\alpha$.
The other inequality is obvious. ∎

######  Remark 2.7.

  1. (1)

The above proof shows that we can indeed assume that either $A_{n}=X$ (if
$V_{n}\subseteq A$), or else $A_{n}=A\cap V_{n}$ (and thus $A_{n}\subseteq
V_{n}$); indeed, if $A$ is not clopen and we drop the requirement that $A_{n}$
be nonselfdual, then the latter can be assumed to be true for all
$n\in\omega$.

  2. (2)

Proposition 2.6 can be relativized to any clopen subset of $X$, namely: If
$A\subseteq X$ is selfdual and $U\in\boldsymbol{\Delta}^{0}_{1}(X)$, then
there exist a clopen partition $(U_{n})_{n\in\omega}$ of $U$ and nonselfdual
sets $A_{n}<_{\mathsf{W}}^{X}A$ such that $\bigcup_{n\in\omega}(A_{n}\cap
U_{n})=A\cap U$. Indeed, this is trivial if $U\cap A<_{\mathsf{W}}^{X}A$. If
instead $U\cap A\equiv_{\mathsf{W}}^{X}A$, then $U\cap A$ is selfdual and we
can apply Proposition 2.5, setting then $U_{n}=V_{n}\cap U$. This easily
provides alternative proofs of [CMM22, Theorem 5.4 and Corollary 5.5].

###  2.3. Pointclasses

Wadge pointclasses provide an alternative (but equivalent) way to present and
study the Wadge hierarchy on a space $X$. A _boldface pointclass_
$\boldsymbol{\Gamma}$ in $X$ is a subset of $\mathscr{P}(X)$ which is closed
under continuous preimages, i.e. it is downward closed under
$\leq_{\mathsf{W}}^{X}$. The _dual_
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}$
of $\boldsymbol{\Gamma}$ is defined by
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}=\\{\neg
A\mid A\in\boldsymbol{\Gamma}\\}$. The dual operator is clearly idempotent,
i.e. the dual of
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}$
is $\boldsymbol{\Gamma}$ itself. We say that $\boldsymbol{\Gamma}$ is
_nonselfdual_ if
$\boldsymbol{\Gamma}\neq\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}$,
and _selfdual_ otherwise.

A boldface pointclass $\boldsymbol{\Gamma}$ is a _Wadge class_ if it is of the
form $A{\downarrow}_{X}=\\{B\subseteq X\mid B\leq_{\mathsf{W}}^{X}A\\}$ for
some $A\subseteq X$; any $A\in\boldsymbol{\Gamma}$ satisfying
$\boldsymbol{\Gamma}=A{\downarrow}_{X}$ is called _complete_ (for
$\boldsymbol{\Gamma}$), and we say that $\boldsymbol{\Gamma}$ is _generated_
by $A$. We denote by $\mathrm{Class}_{\mathsf{W}}(X)$ the collection of all
Wadge classes in $X$. Notice that by $\mathsf{SLO}^{\mathsf{W}}_{X}$, every
nonselfdual boldface pointclass $\boldsymbol{\Gamma}$ of $X$ is automatically
a Wadge class; in contrast, not all selfdual boldface pointclasses in $X$ are
Wadge classes. We denote by $\mathrm{Class}_{\operatorname{NSD}}(X)$ the
collection of all nonselfdual Wadge classes in $X$, and by
$\mathrm{Class}_{\operatorname{SD}}(X)=\mathrm{Class}_{\mathsf{W}}(X)\setminus\mathrm{Class}_{\operatorname{NSD}}(X)$
the collection of the selfdual ones.

The Wadge hierarchy $\mathscr{W}_{X}$ is clearly isomorphic to the structure
of all Wadge classes ordered by inclusion, as witnessed by the map
$[A]_{\mathsf{W}}^{X}\mapsto A{\downarrow}_{X}$. In particular,
$A\in\mathrm{Set}_{\operatorname{SD}}(X)$ if and only if
$A{\downarrow}_{X}\in\mathrm{Class}_{\operatorname{SD}}(X)$, and a nonselfdual
pair $\\{[A]_{\mathsf{W}}^{X},[\neg A]_{\mathsf{W}}^{X}\\}$ corresponds to the
pair of distinct nonselfdual Wadge classes
$(\boldsymbol{\Gamma},\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox})$
with $\boldsymbol{\Gamma}=A{\downarrow}_{X}$. Every
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\mathsf{W}}(X)$ induces a _coarse Wadge
class_
$\boldsymbol{\Gamma}^{*}=\boldsymbol{\Gamma}\cup\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}$.
When $\mathsf{SLO}^{\mathsf{W}}_{X}$ holds and $\leq_{\mathsf{W}}^{X}$ is
well-founded, inclusion on coarse Wadge classes is a well-order: the ordinal
$\alpha$ corresponding to the position of $\boldsymbol{\Gamma}^{*}$ in such a
well-order (once we start counting by $1$) coincides with
$||A||_{\mathsf{W}}^{X}$ for some/any $A\subseteq X$ such that
$\boldsymbol{\Gamma}=A{\downarrow}_{X}$. Moreover, when $X\subseteq Y$ are
zero-dimensional Polish spaces with $X$ closed in $Y$, the retraction method
(Section 2.2) induces an injection from $\mathrm{Class}_{\mathsf{W}}(X)$ into
$\mathrm{Class}_{\mathsf{W}}(Y)$. Indeed, if $\rho\colon Y\to X$ is a
retraction we can associate to each
$\boldsymbol{\Gamma}=A{\downarrow}_{X}\in\mathrm{Class}_{\mathsf{W}}(X)$ the
Wadge class
$\rho^{-1}(\boldsymbol{\Gamma})=(\rho^{-1}(A)){\downarrow}_{Y}\in\mathrm{Class}_{\mathsf{W}}(Y)$.
This is again independent on the chosen retraction, i.e. if
$\rho_{1},\rho_{2}\colon Y\to X$ are both retractions, then
$\rho_{1}^{-1}(\boldsymbol{\Gamma})=\rho_{2}^{-1}(\boldsymbol{\Gamma})$
because $\rho_{1}^{-1}(A)\equiv_{\mathsf{W}}^{Y}\rho_{2}^{-1}(A)$. Clearly,
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(X)\iff\rho^{-1}(\boldsymbol{\Gamma})\in\mathrm{Class}_{\operatorname{NSD}}(Y)$.

Notice that any boldface pointclass $\boldsymbol{\Gamma}\neq\\{\emptyset\\}$
completely determines the space $X$ where it “lives”, as $X$ can be
canonically recovered as the $\subseteq$-largest set in $\boldsymbol{\Gamma}$.
This allows us to greatly simplify the notation: the symbol $\mathsf{W}$ is
unnecessary555The link to continuous reducibility is somewhat implicit in the
fact that we only consider _boldface_ pointclasses, which by definition are
$\leq_{\mathsf{W}}^{X}$-downward closed. because the order relation for
pointclasses is simply inclusion; and as observed we can (almost) always drop
the reference to the ambient space $X$. So for example when
$\boldsymbol{\Gamma}\neq\\{\emptyset\\}$ is a Wadge class we can simply write
$||\boldsymbol{\Gamma}||$ to denote the Wadge rank of $\boldsymbol{\Gamma}$,
i.e. the ordinal $||A||_{\mathsf{W}}^{X}$ for some/any $A\subseteq X$ such
that $\boldsymbol{\Gamma}=A{\downarrow}_{X}$.

The chosen notation highlights the tight connection between ordinals, sets,
and Wadge classes in a zero-dimensional Polish space $X$: if
$\boldsymbol{\Gamma}$ is a Wadge class generated by $A\subseteq X$, then

$\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(X)\iff
A\in\mathrm{Set}_{\operatorname{NSD}}(X)\iff||\boldsymbol{\Gamma}||=||A||_{\mathsf{W}}^{X}\in\mathrm{Ord}_{\mathrm{NSD}}(X).$

###  2.4. The relativization method

Building on previous work by Louveau and Saint-Raymond, the paper [CMM22]
develops a substantial part of Wadge theory for arbitrary zero-dimensional
Polish spaces, often in terms of Wadge classes. Here we recall the few facts
that we need for the present work.

The relativization method, introduced in [LSR88], provides a way to “transfer”
boldface pointclasses in $\mathcal{N}$ to the space $X$ (see [CMM22, Section
6]).

######  Definition 2.8.

Let $X$ be a zero-dimensional Polish space, and consider any boldface
pointclass $\boldsymbol{\Gamma}\subseteq\mathscr{P}(\mathcal{N})$. The
_relativization_ of $\boldsymbol{\Gamma}$ to $X$ is the boldface pointclass

$\boldsymbol{\Gamma}(X)=\\{A\subseteq X\mid
g^{-1}(A)\in\boldsymbol{\Gamma}\text{ for all continuous
}g\colon\mathcal{N}\to X\\}.$

Notice that the dual (in $X$) of $\boldsymbol{\Gamma}(X)$ is simply
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}(X)$,
i.e. the relativization of the dual (in $\mathcal{N}$) of
$\boldsymbol{\Gamma}$. Thus if
$\boldsymbol{\Gamma}(X)\in\mathrm{Class}_{\operatorname{NSD}}(X)$ then
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$. In
contrast, there are situations where
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$ but
$\boldsymbol{\Gamma}(X)\in\mathrm{Class}_{\operatorname{SD}}(X)$, see the end
of this subsection for more details.

The relativization method works particularly well with nonselfdual Wadge
classes. The following result sums up the content of [CMM22, (Proof of) Lemma
6.5 and Theorem 7.2].

######  Theorem 2.9 (${\sf AD}$).

Let $X$ be a zero-dimensional Polish space.

  1. (1)

For all $\mathbf{\Lambda}\in\mathrm{Class}_{\operatorname{NSD}}(X)$ there is a
unique boldface pointclass
$\boldsymbol{\Gamma}\subseteq\mathscr{P}(\mathcal{N})$ such that
$\mathbf{\Lambda}=\boldsymbol{\Gamma}(X)$, and moreover
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$.

  2. (2)

If moreover $X$ is uncountable, then
$\mathrm{Class}_{\operatorname{NSD}}(X)=\\{\boldsymbol{\Gamma}(X)\mid\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})\\}$,
and indeed
$\langle\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N}),{\subseteq}\rangle$
and $\langle\mathrm{Class}_{\operatorname{NSD}}(X),{\subseteq}\rangle$ are
isomorphic via the map $\boldsymbol{\Gamma}\mapsto\boldsymbol{\Gamma}(X)$.

Theorem 2.9(2) relies on the fact that if
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$ then
$\boldsymbol{\Gamma}(X)$ is still nonselfdual if $X$ is uncountable. In
countable spaces this ceases to be true: since all sets in a countable Polish
space $X$ are $\boldsymbol{\Delta}^{0}_{2}$, all pointclasses in
$\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$ containing
$\boldsymbol{\Delta}^{0}_{2}$ relativize to $\mathscr{P}(X)$, which is
selfdual. We will provide the correct analogue of Theorem 2.9(2) for countable
Polish spaces in Corollary 3.19.

###  2.5. Cantor-Bendixson derivatives and rank

Recall from [Kec95, Section 6.C] the classical notion of Cantor-Bendixson
derivative, defined by

$D_{\operatorname{CB}}(Y)=\\{x\in Y\mid x\text{ is not isolated in $Y$}\\},$

and its iterates $D^{\alpha}_{\operatorname{CB}}(X)$, recursively defined by
setting $D^{0}_{\operatorname{CB}}(X)=X$,
$D^{\alpha+1}_{\operatorname{CB}}(X)=D_{\operatorname{CB}}(D^{\alpha}_{\operatorname{CB}}(X))$,
and
$D^{\lambda}_{\operatorname{CB}}(X)=\bigcap_{\alpha<\lambda}D^{\alpha}_{\operatorname{CB}}(X)$
for $\lambda$ limit. When $X$ is Polish, there is $\alpha<\omega_{1}$ such
that $D^{\alpha}_{\operatorname{CB}}(X)=D^{\alpha+1}_{\operatorname{CB}}(X)$:
the smallest such $\alpha$ is the Cantor-Bendixson rank (_CB-rank_ for short)
of $X$ and is denoted by $||X||_{\operatorname{CB}}$, while
$\operatorname{ker}_{\operatorname{CB}}(X)=D^{||X||_{\operatorname{CB}}}_{\operatorname{CB}}(X)$
is called _perfect kernel_ of $X$ and can equivalently be defined as the
largest perfect666A subspace of $X$ is perfect if it is closed and has no
isolated point. subspace of $X$. Moreover, for $x\in
X\setminus\operatorname{ker}_{\operatorname{CB}}(X)$ we let
$||x,X||_{\operatorname{CB}}$ be the unique ordinal
$\alpha<||X||_{\operatorname{CB}}$ satisfying $x\in
D^{\alpha}_{\operatorname{CB}}(X)\setminus
D^{\alpha+1}_{\operatorname{CB}}(X)$.

If $X$ is countable, then
$\operatorname{ker}_{\operatorname{CB}}(X)=\emptyset$. In this case, we define
the _Cantor-Bendixson degree_ $\mathrm{Deg}_{\operatorname{CB}}(X)$ of $X$
(_CB-degree_ for short) as the cardinality of
$D^{\alpha}_{\operatorname{CB}}(X)$ if $||X||_{\operatorname{CB}}=\alpha+1$ is
a successor ordinal, and $\mathrm{Deg}_{\operatorname{CB}}(X)=\omega$
otherwise. The _Cantor-Bendixson type_ of $X$ (_CB-type_ for short) is the
pair
$\operatorname{tp}(X)=(||X||_{\operatorname{CB}},\mathrm{Deg}_{\operatorname{CB}}(X))$.
For compact countable Polish spaces, $\operatorname{tp}(X)$ is a complete
invariant for homeomorphism.

We say that a countable Polish space $X$ is _simple_ if it has successor CB-
rank $||X||_{\operatorname{CB}}=\alpha+1$ and
$\mathrm{Deg}_{\operatorname{CB}}(X)=1$, that is, if
$D^{\alpha}_{\operatorname{CB}}(X)$ is a singleton. The following is a special
case of [Car13, Lemma 2.4], and shows in particular that if $X$ is a countable
Polish space with limit CB-rank, then $X$ is not compact.

######  Lemma 2.10.

Every countable Polish space $X$ of CB-type
$(\alpha,\beta)\in\omega_{1}\times(\omega+1)$ admits a partition in nonempty
clopen subsets $(C_{n})_{n\in\beta}$ such that each $C_{n}$ is simple,
$||C_{n}||_{\operatorname{CB}}=\alpha$ if $\alpha$ is successor, and
$||C_{n}||_{\operatorname{CB}}<\alpha$ with
$\sup_{n\in\omega}||C_{n}||_{\operatorname{CB}}=\alpha$ if $\alpha$ is limit.

The following is instead a direct application of [Car13, Corollary 2.3].

######  Lemma 2.11.

Let $X$ be a zero-dimensional Polish space and $U\subseteq X$ be an open set
such that $U\cap\operatorname{ker}_{\operatorname{CB}}(X)=\emptyset$. Then
$||U||_{\operatorname{CB}}=\min\\{\alpha\leq||X||_{\operatorname{CB}}\mid
U\cap D^{\alpha}_{\operatorname{CB}}(X)=\emptyset\\}$.

###  2.6. The pointed gluing operation

We adopt a standard notation for sequences: for example,
$\operatorname{lh}(s)$ denotes the length of $s$, $s{{}^{\smallfrown}}t$
stands for the concatenation of $s$ and $t$, while for $i\in\omega$ and
$n\in\omega$ we denote by $(i)^{n}$ (respectively, $(i)^{\infty}$) the
constant sequence of length $n$ (respectively, the $\omega$-sequence) with
value $i$. If $X$ is a topological space, we say that a sequence
$(A_{n})_{n\in\omega}$ of subsets of $X$ converges to a point $x\in X$, in
symbols $A_{n}\to x$, if any open neighborhood of $x$ contains all but
finitely many $A_{n}$’s.

The following construction, taken from [Car13, Section 3], is implicitly used
in the area since Wadge’s thesis [Wad83] and has been considered, sometimes
with slight variations, by several authors (see e.g. [Sel04, Section 5]).
Given a sequence $(A_{n})_{n\in\omega}$ with $A_{n}\subseteq\mathcal{N}$, its
_pointed gluing_ is the set

$\operatorname{ptgl}((A_{n})_{n\in\omega})=(0)^{\infty}\cup\bigcup_{n\in\omega}\\{{(0)^{n}}{{}^{\smallfrown}}(1){{}^{\smallfrown}}x\mid
x\in A_{n}\\}$

Intuitively, $\operatorname{ptgl}((A_{n})_{n\in\omega})$ is thus obtained by
taking copies of the $A_{n}$’s which converge to $(0)^{\infty}$, together with
such limit. The functional counterpart of the pointed gluing operation is
defined by considering a sequence $(f_{n})_{n\in\omega}$ of functions
$f_{n}\colon A_{n}\to B_{n}$, and letting

$\operatorname{ptgl}((f_{n})_{n\in\omega})\colon\operatorname{ptgl}((A_{n})_{n\in\omega})\to\operatorname{ptgl}((B_{n})_{n\in\omega})$

be defined as the map sending $(0)^{\infty}$ to itself and points of the form
$(0)^{n}{{}^{\smallfrown}}(1){{}^{\smallfrown}}x$ with $x\in A_{n}$ to
$(0)^{n}{{}^{\smallfrown}}(1){{}^{\smallfrown}}f_{n}(x)$. When $A_{n}=A$ or
$f_{n}=f$ for all $n\in\omega$, to simplify the notation we write
$\operatorname{ptgl}(A)$ instead of
$\operatorname{ptgl}((A_{n})_{n\in\omega})$ or $\operatorname{ptgl}(f)$
instead of $\operatorname{ptgl}((f_{n})_{n\in\omega})$, respectively.

It is easy to verify that if $A_{n}$ is closed (respectively, compact), then
so is $\operatorname{ptgl}((A_{n})_{n\in\omega})$, and that
$\operatorname{ptgl}((f_{n})_{n\in\omega})$ is continuous (respectively,
injective, an embedding, or a homeomorphism) if so are the $f_{n}$’s.
Moreover, if the sequence of ordinals
$(||A_{n}||_{\operatorname{CB}})_{n\in\omega}$ is non-decreasing, then an easy
computation shows that
$||\operatorname{ptgl}((A_{n})_{n\in\omega})||_{\operatorname{CB}}=(\sup_{n\in\omega}||A_{n}||_{\operatorname{CB}})+1$
(see e.g. [Car13, Proposition 3.1]).

###  2.7. Difference hierarchy

We recall the definition of difference hierarchy (see e.g. [Kec95, Section
22.E]). Let $X$ be a Polish space, $\alpha\geq 1$ be a countable ordinal, and
$(A_{\gamma})_{\gamma<\alpha}$ be a sequence of subsets of $X$. Then we define

$D_{\alpha}((A_{\gamma})_{\gamma<\alpha})=\bigcup\bigg{\\{}A_{\gamma}\setminus\bigcup_{\gamma^{\prime}<\gamma}A_{\gamma^{\prime}}\mid\gamma<\alpha\text{
and }\gamma\text{ has parity opposite to }\alpha\bigg{\\}}.$

Moreover, when the boldface pointclass
$\boldsymbol{\Gamma}\subseteq\mathscr{P}(X)$ is different from
$\\{\emptyset\\}$ we set
$D_{\alpha}(\boldsymbol{\Gamma})=\\{D_{\alpha}((A_{\gamma})_{\gamma<\alpha})\mid
A_{\gamma}\in\boldsymbol{\Gamma}\\}$. By convention, we set
$D_{0}(\boldsymbol{\Gamma})=\\{\emptyset\\}$. Notice also that
$D_{1}(\boldsymbol{\Gamma})=\boldsymbol{\Gamma}$.

We will always work with $\boldsymbol{\Gamma}=\boldsymbol{\Sigma}^{0}_{1}(X)$.
One can easily verify that $D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$ is a
boldface pointclass (in $X$), and that
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))=D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1})(X)$.
Thus we can unambiguously denote by
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$
the dual of $D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$.

Wadge himself proved in [Wad83] that the first $\omega_{1}$-many nonselfdual
levels of $\mathscr{W}_{\mathcal{N}}$ are occupied by the difference classes
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1})$, $\alpha<\omega_{1}$, and their
duals. It follows that
$||D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1})||_{\mathcal{N}}=2\alpha+1$, and
that using the well-known successor and countable sum operations from [Wad83,
Chapter 3] one can define complete sets (up to Wadge equivalence) for such
classes. More precisely, we recursively define a sequence of Wadge degrees
$(\mathbf{D}_{\alpha})_{\alpha<\omega_{1}}$ by setting

(2.1)
$\mathbf{D}_{\alpha}=\begin{cases}[\emptyset]_{\mathsf{W}}^{\mathcal{N}}&\text{if
}\alpha=0\\\ [\\{(0)^{n}{{}^{\smallfrown}}(m+1){{}^{\smallfrown}}x\mid
n,m\in\omega\wedge x\in\neg B_{\beta}\\}]_{\mathsf{W}}^{\mathcal{N}}&\text{if
}\alpha=\beta+1\\\ [\\{(0)^{n}{{}^{\smallfrown}}(m+1){{}^{\smallfrown}}x\mid
n,m\in\omega\wedge x\in\neg
B_{\alpha_{n}}\\}]_{\mathsf{W}}^{\mathcal{N}}&\text{if }\alpha\text{ is
limit,}\end{cases}$

where $B_{\beta}$ is any set in $\mathbf{D}_{\beta}$,
$(\alpha_{n})_{n\in\omega}$ is some/any increasing sequence cofinal in
$\alpha$, and $B_{\alpha_{n}}$ is an arbitrary set in
$\mathbf{D}_{\alpha_{n}}$. An easy induction shows that the sequence
$(\mathbf{D}_{\alpha})_{\alpha<\omega_{1}}$ is well-defined, i.e. that up to
$\equiv_{\mathsf{W}}^{\mathcal{N}}$ it is independent of the choice of
$B_{\beta}$, $(\alpha_{n})_{n\in\omega}$, and $B_{\alpha_{n}}$. Moreover, one
can prove by induction again that if $B_{\alpha}\in\mathbf{D}_{\alpha}$ then
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1})=(B_{\alpha}){\downarrow}_{\mathcal{N}}$.

##  3\. Our toolbox

###  3.1. More on the retraction method

The retraction method immediately settles our problem for zero-dimensional
Polish spaces which are not _$\sigma$ -compact_, i.e. cannot be written as a
countable union of compact spaces, and for those which are uncountable and
compact. This shows that we could in principle concentrate on $\sigma$-compact
non-compact spaces, although our analysis will be more general and work for
all zero-dimensional Polish spaces, without those limitations.

The next two results do not require any extra assumption beyond our base
theory ${\sf ZF}+{\sf DC}(\mathbb{R})$.

######  Lemma 3.1.

Suppose that $X,Y$ are zero-dimensional Polish spaces, each of which is
homeomorphic to closed subset of the other one. Then $\mathscr{W}_{X}$ and
$\mathscr{W}_{Y}$ are isomorphic.

###### Proof.

Without loss of generality, $X$ is a closed subset of $Y$ and $Y$ is a closed
subspace of some $X^{\prime}$ which is homeomorphic to $X$ via some $h\colon
X\to X^{\prime}$. Let $\rho_{1}\colon Y\to X$ and $\rho_{2}\colon
X^{\prime}\to Y$ be retractions. We already observed that
$A\mapsto\rho_{1}^{-1}(A)$ provides an embedding of
$\langle\mathscr{P}(X),{\leq_{\mathsf{W}}^{X}},{\neg}\rangle$ into
$\langle\mathscr{P}(Y),{\leq_{\mathsf{W}}^{Y}},{\neg}\rangle$, and hence it
canonically induces an embedding of $\mathscr{W}_{X}$ into $\mathscr{W}_{Y}$:
we claim that the latter is also surjective (hence an isomorphism), i.e. that
for every $B\subseteq Y$ there is $A\subseteq X$ such that
$B\equiv_{\mathsf{W}}^{Y}\rho_{1}^{-1}(A)$. To see this, it is enough to
notice that since $\rho_{1}$ and $\rho_{2}$ are retractions and $h$ is a
homeomorphism, then

$B\equiv_{\mathsf{W}}^{Y,X^{\prime}}\rho_{2}^{-1}(B)\equiv_{\mathsf{W}}^{X^{\prime},X}h^{-1}(\rho_{2}^{-1}(B))\equiv_{\mathsf{W}}^{X,Y}\rho_{1}^{-1}(h^{-1}(\rho_{2}^{-1}(B))).$

Thus setting $A=h^{-1}(\rho_{2}^{-1}(B))$ we are done. ∎

Combining the above lemma with the universality properties of $\mathcal{N}$
and $\mathcal{C}$ ([Kec95, Theorem 7.8]), the Cantor-Bendixson theorem
([Kec95, Corollary 6.5]) and the Hurewicz theorem ([Kec95, Theorem 7.10]) we
obtain the following.

######  Proposition 3.2.

Let $X$ be a zero-dimensional Polish space.

  1. (1)

If $X$ is not $\sigma$-compact,777Notice that this hypothesis implies in
particular that $X$ is uncountable. then $\mathscr{W}_{X}$ is isomorphic to
$\mathscr{W}_{\mathcal{N}}$.

  2. (2)

If $X$ is compact and uncountable, then $\mathscr{W}_{X}$ is isomorphic to
$\mathscr{W}_{\mathcal{C}}$.

######  Remark 3.3.

Since $\sigma$-compactness already appeared in [MRSS15, Theorem 5.12(2)] as a
dividing line for Wadge-like hierarchies on zero-dimensional Polish space, one
may be tempted to conjecture that Proposition 3.2(1) can be turned into a
characterization. We will however show that this is not the case: by 1, there
are $\sigma$-compact spaces whose Wadge hierarchy is isomorphic to
$\mathscr{W}_{\mathcal{N}}$ (take any $\sigma$-compact space with non-compact
perfect kernel, e.g. $\omega\times\mathcal{C}$), and there are non-compact
spaces whose Wadge hierarchy is isomorphic to $\mathscr{W}_{\mathcal{C}}$
(take any non-compact space with compact kernel and finite CB-rank, e.g.
$\omega\oplus\mathcal{C}$ where $\oplus$ denotes the disjoint sum and $\omega$
is discrete).

###  3.2. More on the relativization method

The next lemma shows that for any retraction $\rho\colon\mathcal{N}\to X$, the
map $\boldsymbol{\Gamma}\mapsto\rho^{-1}(\boldsymbol{\Gamma})$ from
$\mathrm{Class}_{\mathsf{W}}(X)$ to $\mathrm{Class}_{\mathsf{W}}(\mathcal{N})$
is a right inverse of (the restriction to
$\mathrm{Class}_{\mathsf{W}}(\mathcal{N})$ of) the relativization map
$\boldsymbol{\Gamma}\mapsto\boldsymbol{\Gamma}(X)$. The proof is the same as
the first part of the proof of [CMM22, Lemma 6.5] and does not require extra
assumptions beyond ${\sf ZF}+{\sf DC}(\mathbb{R})$.

######  Lemma 3.4.

Let $X\subseteq\mathcal{N}$ be closed, and $\rho\colon\mathcal{N}\to X$ be a
retraction. Then for every $\mathbf{\Lambda}\in\mathrm{Class}_{\mathsf{W}}(X)$

$(\rho^{-1}(\mathbf{\Lambda}))(X)=\mathbf{\Lambda}.$

###### Proof.

Let $A\subseteq X$ be complete for $\mathbf{\Lambda}$. If
$B\in(\rho^{-1}(\mathbf{\Lambda}))(X)$ then
$\rho^{-1}(B)\in\rho^{-1}(\mathbf{\Lambda})$ because $\rho$ is continuous,
thus $\rho^{-1}(B)\leq_{\mathsf{W}}^{\mathcal{N}}\rho^{-1}(A)$ by definition
of $\rho^{-1}(\mathbf{\Lambda})$, hence $B\leq_{\mathsf{W}}^{X}A$, and finally
$B\in\mathbf{\Lambda}$ by choice of $A$.

Conversely, pick any $B\in\mathbf{\Lambda}$, let $f\colon X\to X$ witness
$B\leq_{\mathsf{W}}^{X}A$, and let $g\colon\mathcal{N}\to X$ be any continuous
function. Then $f\circ g$ witnesses
$g^{-1}(B)\leq_{\mathsf{W}}^{\mathcal{N},X}A$, and since
$A\equiv_{\mathsf{W}}^{X,\mathcal{N}}\rho^{-1}(A)$ we get
$g^{-1}(B)\leq_{\mathsf{W}}^{\mathcal{N}}\rho^{-1}(A)$, i.e.
$g^{-1}(B)\in\rho^{-1}(\mathbf{\Lambda})$. Since $g$ was arbitrary, this shows
that $B\in(\rho^{-1}(\mathbf{\Lambda}))(X)$, as desired. ∎

######  Corollary 3.5 (${\sf AD}$).

Let $X\subseteq Y$ be zero-dimensional Polish spaces with $X$ closed in $Y$,
and let $\boldsymbol{\Gamma}\subseteq\mathscr{P}(\mathcal{N})$ be a boldface
pointclass. If
$\boldsymbol{\Gamma}(X)\in\mathrm{Class}_{\operatorname{NSD}}(X)$, then
$\boldsymbol{\Gamma}(Y)\in\mathrm{Class}_{\operatorname{NSD}}(Y)$. Moreover,
if $A\subseteq X$ is complete for $\boldsymbol{\Gamma}(X)$ and $\rho\colon
Y\to X$ is any retraction, then $\rho^{-1}(A)$ is complete for
$\boldsymbol{\Gamma}(Y)$.

###### Proof.

Without loss of generality, we may assume that $Y$ (and hence also $X$) is a
closed subset of $\mathcal{N}$. Let $A$ and $\rho$ be as in the statement, and
fix a retraction $\rho^{\prime}\colon\mathcal{N}\to Y$, so that
$\rho\circ\rho^{\prime}$ is a retraction of $\mathcal{N}$ onto $X$. Then
$(\rho\circ\rho^{\prime})^{-1}(\boldsymbol{\Gamma}(X))=\boldsymbol{\Gamma}$ by
Lemma 3.4 applied to $\mathbf{\Lambda}=\boldsymbol{\Gamma}(X)$ and Theorem
2.9(1). Using this and applying once again Lemma 3.4 with
$\mathbf{\Lambda}=\rho^{-1}(\boldsymbol{\Gamma}(X))\in\mathrm{Class}_{\mathsf{W}}(Y)$,
it follows that

$\rho^{-1}(\boldsymbol{\Gamma}(X))=((\rho^{\prime})^{-1}(\rho^{-1}(\boldsymbol{\Gamma}(X))))(Y)=((\rho\circ\rho^{\prime})^{-1}(\boldsymbol{\Gamma}(X)))(Y)=\boldsymbol{\Gamma}(Y),$

which proves at once that $\boldsymbol{\Gamma}(Y)$ is nonselfdual (because so
is $\rho^{-1}(\boldsymbol{\Gamma}(X))$) and that $\rho^{-1}(A)$ is complete
for $\boldsymbol{\Gamma}(Y)$. ∎

The relativization method and, in particular, Corollary 3.5, gives a general
version of the classical computation in $\mathcal{N}$ of the supremum of at
most countably many nonselfdual classes, that we use several times in this
paper.

######  Lemma 3.6 (${\sf AD}$).

Let $X$ be a zero-dimensional Polish space, $(X_{i})_{i\in I}$ be a clopen
partition of $X$ for some $I\subseteq\omega$, and
$(\boldsymbol{\Gamma}_{i})_{i\in I}$ be boldface pointclasses in
$\mathcal{N}$. Suppose that
$\boldsymbol{\Gamma}_{i}(X_{i})\in\mathrm{Class}_{\operatorname{NSD}}(X_{i})$
for all $i\in I$, and let $A_{i}\subseteq X_{i}$ be complete for
$\boldsymbol{\Gamma}_{i}(X_{i})$. Then the pointclass
$\boldsymbol{\Gamma}=A{\downarrow}_{X}\in\mathrm{Class}_{\mathsf{W}}(X)$
generated by $A=\bigcup_{i\in I}A_{i}$ is the smallest Wadge class such that
$\boldsymbol{\Gamma}\supseteq\boldsymbol{\Gamma}_{i}(X)$ for all $i\in I$.

Moreover, if for every $i\in I$ there is $j\in I$ such that
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}_{i}\subseteq\boldsymbol{\Gamma}_{j}$,
then $\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{SD}}(X)$.

The class $\boldsymbol{\Gamma}=A{\downarrow}_{X}$ from the lemma is called the
_supremum_ of the $\boldsymbol{\Gamma}_{i}$’s in $X$.

###### Proof.

For each $i\in I$ pick a retraction $\rho_{i}\colon X\to X_{i}$, so that
$\boldsymbol{\Gamma}_{i}(X)=\rho_{i}^{-1}(A_{i}){\downarrow}_{X}$ by Corollary
3.5. Since the identity function witnesses $A_{i}\leq_{\mathsf{W}}^{X_{i},X}A$
and $\rho_{i}^{-1}(A_{i})\equiv_{\mathsf{W}}^{X,X_{i}}A_{i}$, we get
$\boldsymbol{\Gamma}_{i}(X)\subseteq A{\downarrow}_{X}=\boldsymbol{\Gamma}$.
Moreover, if $B\subseteq X$ is such that
$B{\downarrow}_{X}\supseteq\boldsymbol{\Gamma}_{i}(X)$, then
$\rho_{i}^{-1}(A_{i})\leq_{\mathsf{W}}^{X}B$, and hence
$A_{i}\leq_{\mathsf{W}}^{X_{i},X}B$ via some $f_{i}\colon X_{i}\to X$. It
follows that $\bigcup_{i\in I}f_{i}$ witnesses $A\leq_{\mathsf{W}}^{X}B$, and
thus $\boldsymbol{\Gamma}\subseteq B{\downarrow}_{X}$.

For the second part, notice that we can assume without loss of generality that
$X$ is closed in $\mathcal{N}$, so that the $X_{i}$’s are closed in
$\mathcal{N}$ as well. Let $r_{i}\colon\mathcal{N}\to X_{i}$ be retractions.
If $i,j\in I$ are as in the statement, then using Corollary 3.5 again we get

$X_{i}\setminus
A_{i}\equiv_{\mathsf{W}}^{X_{i},\mathcal{N}}{\mathcal{N}\setminus
r_{i}^{-1}(A_{i})}\leq_{\mathsf{W}}^{\mathcal{N}}r_{j}^{-1}(A_{j})\equiv_{\mathsf{W}}^{\mathcal{N},X_{j}}A_{j},$

where the middle reduction exists because $\mathcal{N}\setminus
r_{i}^{-1}(A_{i})\in\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}_{i}\subseteq\boldsymbol{\Gamma}_{j}$.
If $f_{i}\colon X_{i}\to X_{j}$ witnesses $X_{i}\setminus
A_{i}\leq_{\mathsf{W}}^{X_{i},X_{j}}A_{j}$, then $\bigcup_{i\in I}f_{i}$
witnesses $\neg A\leq_{\mathsf{W}}^{X}A$, as desired. ∎

The typical situations in which we will apply the “moreover” part of Lemma 3.6
are when $I=2$ and
$\boldsymbol{\Gamma}_{0}=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}_{1}$,
or $I=\omega$ and $\boldsymbol{\Gamma}_{i}\subsetneq\boldsymbol{\Gamma}_{i+1}$
(in the second case use Wadge’s Lemma to obtain
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}_{i}\subseteq\boldsymbol{\Gamma}_{i+1}$).
Notice also that if $X$ is uncountable, then by Theorem 2.9(2) the hypothesis
$\boldsymbol{\Gamma}_{i}(X)\in\mathrm{Class}_{\operatorname{NSD}}(X_{i})$
follows from the more manageable
$\boldsymbol{\Gamma}_{i}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$.

###  3.3. Compact rank

An important dividing line in our analysis is whether the given Polish space
$X$ has a compact perfect kernel or not. If $X$ has a compact perfect kernel,
then its iterated Cantor-Bendixson derivatives are eventually compact. We use
this idea to define a new ordinal invariant, called compact rank.

######  Definition 3.7.

Let $X$ be a Polish space such that
$\operatorname{ker}_{\operatorname{CB}}(X)$ is compact. Then the _compact
rank_ of $X$ is

$||X||_{\operatorname{Comp}}=\min\\{\alpha\leq||X||_{\operatorname{CB}}\mid
D^{\alpha}_{\operatorname{CB}}(X)\text{ is compact}\\}.$

The assumption on $\operatorname{ker}_{\operatorname{CB}}(X)$ ensures that
$||X||_{\operatorname{Comp}}$ is well-defined. Moreover,
$||X||_{\operatorname{Comp}}=0$ if and only if the whole $X$ is compact.

There is a natural characterization of when a space with compact perfect
kernel is not compact itself which will be useful later on. It is based on the
following well-known exercise.888The nontrivial direction follows from the
fact that if $\\{U_{n}\mid n\in\omega\\}$ is a countable clopen cover of the
space which does not have a finite subcover, then setting
$V_{n}=U_{n}\setminus\bigcup_{j<n}U_{j}$ we get that $\\{V_{n}\mid
n\in\omega\wedge V_{n}\neq\emptyset\\}$ is an infinite clopen partition of the
space.

######  Fact 3.8.

A zero-dimensional second-countable space is not compact if and only if it
admits an infinite (countable) clopen partition.

######  Lemma 3.9.

Suppose that $X$ is a zero-dimensional Polish space with
$\operatorname{ker}_{\operatorname{CB}}(X)$ compact. Then $X$ is not compact
if and only if there exists an infinite discrete clopen set $U\subseteq X$.

###### Proof.

If $U$ is infinite discrete then it is not compact, and if it is closed in $X$
then $X$ cannot be compact either.

Conversely, suppose that $X$ is not compact. By 3.8, there exists an infinite
clopen partition $(C_{n})_{n\in\omega}$ of $X$. Since
$\operatorname{ker}_{\operatorname{CB}}(Z)$ is compact, there exists
$m\in\omega$ such that
$\operatorname{ker}_{\operatorname{CB}}(Z)\subseteq\bigcup_{n\leq m}C_{n}$.
For each $n>m$ there is at least one isolated point $x_{n}\in C_{n}$ because
$C_{n}\cap\operatorname{ker}_{\operatorname{CB}}(X)=\emptyset$: then
$U=\\{x_{n}\mid n>m\\}$ is as required. ∎

It might be interesting to notice that the compact rank can be obtained
through a corresponding derivative $D_{\operatorname{Comp}}(F)$ defined by

$D_{\operatorname{Comp}}(F)=F\setminus\bigcup\\{U\mid U\text{ is infinite
discrete clopen in }F\\}.$

(With this terminology, Lemma 3.9 can be reformulated as: If
$\operatorname{ker}_{\operatorname{CB}}(X)$ is compact, then $X$ is compact if
and only if $D_{\operatorname{Comp}}(X)=X$.) One can then prove that if $X$ is
a non-compact zero-dimensional Polish space with
$\operatorname{ker}_{\operatorname{CB}}(X)$ compact, then
$D_{\operatorname{CB}}(X)=D_{\operatorname{Comp}}(X)$, and thus
$||X||_{\operatorname{Comp}}$ is the rank associated to the derivative
$D_{\operatorname{Comp}}$, i.e.
$||X||_{\operatorname{Comp}}=\min\\{\alpha<\omega_{1}\mid
D^{\alpha}_{\operatorname{Comp}}(X)=D^{\alpha+1}_{\operatorname{Comp}}(X)\\}.$

We will also need the following technical lemma.

######  Lemma 3.10.

Suppose that $X$ is a non-compact zero-dimensional Polish space with
$\operatorname{ker}_{\operatorname{CB}}(X)$ compact. Then for all
$\alpha<||X||_{\operatorname{Comp}}$, there exists an infinite clopen
partition $(C_{n})_{n\in\omega}$ of $X$ such that $C_{0}\supseteq
D^{\alpha+1}_{\operatorname{CB}}(X)$ and
$||C_{n+1}||_{\operatorname{CB}}=\alpha+1$ for all $n\in\omega$.

###### Proof.

Since $\alpha<||X||_{\operatorname{Comp}}\leq||X||_{\operatorname{CB}}$ we
have $D^{\alpha+1}_{\operatorname{CB}}(X)\subsetneq
D^{\alpha}_{\operatorname{CB}}(X)$ and the latter is not compact. By Lemma 3.9
there is an infinite discrete clopen (relatively to
$D^{\alpha}_{\operatorname{CB}}(X)$) set $U=\\{x_{i}\mid
i\in\omega\\}\subseteq D^{\alpha}_{\operatorname{CB}}(X)$, so that $U\cap
D^{\alpha+1}_{\operatorname{CB}}(X)=\emptyset$. Let $(U_{n})_{n\in\omega}$ be
open sets of $X$ such that $U_{0}\cap
D^{\alpha}_{\operatorname{CB}}(X)=D^{\alpha}_{\operatorname{CB}}(X)\setminus
U$ and $U_{i+1}\cap D^{\alpha}_{\operatorname{CB}}(X)=\\{x_{i}\\}$. Without
loss of generality, we may assume that $\bigcup_{n\in\omega}U_{n}=X$
(otherwise we replace $U_{0}$ with $U_{0}\cup(X\setminus
D^{\alpha}_{\operatorname{CB}}(X)$). Using the generalized reduction property
([Kec95, Theorem 22.16]) we obtain a clopen partition $(C_{n})_{n\in\omega}$
of $X$ such that $C_{n}\subseteq U_{n}$, so that in particular $C_{n}\cap
D^{\alpha}_{\operatorname{CB}}(X)=U_{n}\cap D^{\alpha}_{\operatorname{CB}}(X)$
because $(U_{n}\cap D^{\alpha}_{\operatorname{CB}}(X))_{n\in\omega}$ is a
clopen partition of $D^{\alpha}_{\operatorname{CB}}(X)$. Since $U\cap
D^{\alpha+1}_{\operatorname{CB}}(X)=\emptyset$, by Lemma 2.11 the clopen
partition $(C_{n})_{n\in\omega}$ of $X$ is as desired. ∎

###  3.4. Minimal countable spaces

We define (up to homeomorphism) a canonical sequence
$(K_{\alpha+1})_{\alpha<\omega_{1}}$ of countable compact metrizable spaces
such that each $K_{\alpha+1}$ has CB-type $(\alpha+1,1)$ (thus it is simple)
and embeds into every Polish space of CB-rank at least $\alpha+1$.

######  Definition 3.11.

Set $K_{0}=\emptyset$ and inductively define the space $K_{\alpha+1}$ for all
$\alpha<\omega_{1}$ as follows:

  * •

$K_{\alpha+1}=\operatorname{ptgl}(K_{\alpha})$, when $\alpha=0$ or $\alpha$ is
a successor ordinal;

  * •

$K_{\alpha+1}=\operatorname{ptgl}((K_{\alpha_{n}+1})_{n\in\omega})$ for some
$(\alpha_{n})_{n\in\omega}$ cofinal in $\alpha$, when $\alpha$ is limit.

Note that $K_{1}=\\{(0)^{\infty}\\}$. By induction on $\alpha<\omega_{1}$, one
easily sees that $K_{\alpha+1}$ is compact and
$\operatorname{tp}(K_{\alpha+1})=(\alpha+1,1)$. As a consequence, since the
CB-type is a complete invariant for compact countable Polish spaces the
definition of $K_{\alpha+1}$ when $\alpha$ is limit does not depend on the
choice of the cofinal sequence $(\alpha_{n})_{n\in\omega}$, so the sequence is
well-defined up to homeomorphism.999Recall that in models of ${\sf AD}$, it is
not possible to define an $\omega_{1}$-sequence of distinct compact sets
because, by the Perfect Set Property (see e.g. [Jec03, Theorem 33.3]), no
uncountable subset of a standard Borel space can be well-orderable: this is
why we insist here that the definition is given only up to homeomorphism. What
we will use is just the existence, for any fixed $\alpha<\omega_{1}$, of a
countable compact space $K_{\alpha+1}$ with CB-type $(\alpha+1,1)$ and the way
it can be constructed from analogous spaces of lower CB-rank, but its actual
presentation as a subspace of $\mathcal{N}$ will be totally irrelevant.

######  Proposition 3.12.

For every ordinal $\alpha<\omega_{1}$ and every Polish space $X$ with
$||X||_{\operatorname{CB}}\geq\alpha+1$ there exists an embedding $i\colon
K_{\alpha+1}\to X\setminus\operatorname{ker}_{\operatorname{CB}}(X)$ with
compact (hence closed) range.

###### Proof.

Since $K_{\alpha+1}$ is compact, it is enough to prove by induction on
$\alpha$ that there is a continuous injection $i\colon K_{\alpha+1}\to
X\setminus\operatorname{ker}_{\operatorname{CB}}(X)$.

If $\alpha=0$ then $K_{1}=\\{(0)^{\infty}\\}$ and $X$ has at least one
isolated point $x$: setting $i((0)^{\infty})=x$ we are done.

Assume now that $\alpha>0$. Since $||X||_{\operatorname{CB}}\geq\alpha+1$,
then $D^{\alpha}_{\operatorname{CB}}(X)\setminus
D^{\alpha+1}_{\operatorname{CB}}(X)\neq\emptyset$. Pick any $x\in
D^{\alpha}_{\operatorname{CB}}(X)\setminus
D^{\alpha+1}_{\operatorname{CB}}(X)$ and fix any $\subseteq$-decreasing clopen
basis $(V_{n})_{n\in\omega}$ of neighborhoods of $x$ with $V_{0}\cap
D^{\alpha}_{\operatorname{CB}}(X)=\\{x\\}$ (hence also $V_{0}\cap
D^{\alpha+1}_{\operatorname{CB}}(X)=\emptyset$). Set $U_{n}=V_{n}\setminus
V_{n+1}$, so that each $U_{n}$ is a countable clopen set, $U_{n}\to x$,
$||U_{n}||_{\operatorname{CB}}\leq\alpha$ because $U_{n}\cap
D^{\alpha}_{\operatorname{CB}}(X)=\emptyset$, and
$\limsup_{n\in\omega}||U_{n}||_{\operatorname{CB}}=\alpha$101010That is,
$\forall\beta<\alpha\forall n\in\omega\exists m\geq
n\,(||U_{n}||_{\operatorname{CB}}>\beta)$. because every clopen $x\in
V\subseteq V_{0}$ is such that $||V||_{\operatorname{CB}}=\alpha+1$. (Here we
repeatedly use Lemma 2.11.) Let $(\alpha_{n})_{n\in\omega}$ be the constant
sequence with value $\alpha$ if the latter is a successor ordinal, and a
sequence of successor ordinals cofinal in $\alpha$ otherwise. By
$\limsup_{n\in\omega}||U_{n}||_{\operatorname{CB}}=\alpha$, there is an
injective $f\colon\omega\to\omega$ such that
$||U_{f(n)}||_{\operatorname{CB}}\geq\alpha_{n}$ for all $n\in\omega$, so that
by inductive hypothesis there is a continuous injection $i_{n}\colon
K_{\alpha_{n}}\to U_{f(n)}$. Then the map $i\colon K_{\alpha+1}\to
V_{0}\subseteq X\setminus\operatorname{ker}_{\operatorname{CB}}(X)$ defined by

$i(y)=\begin{cases}x&\text{if }y=(0)^{\infty}\\\ i_{n}(z)&\text{if
}y=(0)^{n}{}^{\smallfrown}{}(1){}^{\smallfrown}{}z\text{ with }z\in
K_{\alpha_{n}}\end{cases}$

is the desired continuous injection. ∎

###  3.5. More on the difference hierarchy

Section 2.7 and Theorem 2.9 yields a description of the first
$\omega_{1}$-many nonselfdual Wadge classes in $\mathscr{W}_{X}$ when $X$ is
uncountable. (Here we are not requiring ${\sf AD}$ because Borel determinacy
suffices.)

######  Lemma 3.13.

Let $X$ be an uncountable zero-dimensional Polish space and
$\alpha<\omega_{1}$. Then the $\alpha$-th nonselfdual ordinal in $X$ is
precisely the Wadge rank of $D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$.

Moving to countable Polish spaces $X$, we instead observe that the difference
hierarchy trivializes at least from $||X||_{\operatorname{CB}}$ on. Recall
that we only consider nonempty spaces $X$.

######  Proposition 3.14.

Let $X$ be a countable Polish space. If $||X||_{\operatorname{CB}}\leq\alpha$
then $\mathscr{P}(X)=D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$, and thus
$\mathscr{P}(X)=D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))\cap\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$.
Moreover, if $X$ is simple and $||X||_{\operatorname{CB}}=\beta+1$, then
$\mathscr{P}(X)=D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))\cup\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$.

###### Proof.

Since $D_{\gamma}(\boldsymbol{\Sigma}^{0}_{1}(X))\subseteq
D_{\gamma^{\prime}}(\boldsymbol{\Sigma}^{0}_{1}(X))$ if
$\gamma\leq\gamma^{\prime}$, it is enough to consider the case
$||X||_{\operatorname{CB}}=\alpha$ and prove by induction that
$\mathscr{P}(X)=D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$. Note that
$1\leq\alpha<\omega_{1}$ and $D^{\alpha}_{\operatorname{CB}}(X)=\emptyset$
because $X\neq\emptyset$ is countable.

If $\alpha=1$ then $X$ is discrete, so
$\mathscr{P}(X)=\boldsymbol{\Sigma}^{0}_{1}(X)=D_{1}(\boldsymbol{\Sigma}^{0}_{1}(X))$.
If moreover $X$ is simple, then $X$ is a singleton and
$\mathscr{P}(X)=\\{\emptyset,X\\}=D_{0}(\boldsymbol{\Sigma}^{0}_{1}(X))\cup\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{0}(\boldsymbol{\Sigma}^{0}_{1}(X))$.

Suppose now that $\alpha=\beta+1$ for some $1\leq\beta<\omega_{1}$, and
consider an arbitrary $A\subseteq X$. Let $Y=X\setminus
D^{\beta}_{\operatorname{CB}}(X)$ and $A^{\prime}=Y\setminus A$. Since
$||Y||_{\operatorname{CB}}=\beta$ by Lemma 2.11, we get $A^{\prime}\in
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(Y))\subseteq
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$ by inductive hypothesis and the
fact that $Y$ is open in $X$, so we can write
$A^{\prime}=D_{\beta}((U_{\gamma})_{\gamma<\beta})$ with
$U_{\gamma}\in\boldsymbol{\Sigma}^{0}_{1}(X)$. Moreover,
$D^{\beta}_{\operatorname{CB}}(X)$ is discrete because
$||X||_{\operatorname{CB}}=\beta+1$, therefore $A\cap
D^{\beta}_{\operatorname{CB}}(X)$ is open in
$D^{\beta}_{\operatorname{CB}}(X)$: let $U\in\boldsymbol{\Sigma}^{0}_{1}(X)$
be such that $U\cap D^{\beta}_{\operatorname{CB}}(X)=A\cap
D^{\beta}_{\operatorname{CB}}(X)$, so that the open subset $U_{\beta}=U\cup Y$
of $X$ satisfies $A=U_{\beta}\setminus A^{\prime}$: then
$A=D_{\beta+1}((U_{\gamma})_{\gamma<\beta+1})$, witnessing $A\in
D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$. If moreover $X$ is simple, then
$D^{\beta}_{\operatorname{CB}}(X)=\\{x\\}$ is a singleton. If $x\notin A$,
then $A\subseteq Y$ and using the induction hypothesis we again obtain $A\in
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(Y))\subseteq
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$. If instead $x\in A$, then
$A=X\setminus A^{\prime}$, hence
$A\in\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$.
In both cases, $A\in
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))\cup\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$.

Finally, suppose that $\alpha$ is limit, and take a strictly increasing
sequence $(\alpha_{n})_{n\in\omega}$ of nonzero ordinals cofinal in $\alpha$.
For $n\in\omega$, consider the open sets $V_{n}^{*}=X\setminus
D^{\alpha_{n}}_{\operatorname{CB}}(X)$. Apply the generalized reduction
property to the sequence $(V_{n}^{*})_{n\in\omega}$ to obtain a clopen
partition $(V_{n})_{n\in\omega}$ of $X$ such that $V_{n}\subseteq V_{n}^{*}$.
Given any $A\subseteq X$, set $A_{n}=A\cap V_{n}$. Then $A_{n}\subseteq V_{n}$
and $||V_{n}||_{\operatorname{CB}}\leq\alpha_{n}$ by Lemma 2.11, so by
induction hypothesis $A_{n}\in
D_{\alpha_{n}}(\boldsymbol{\Sigma}^{0}_{1}(V_{n}))\subseteq
D_{\alpha_{n}}(\boldsymbol{\Sigma}^{0}_{1}(X))\subseteq
D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$. Write each $A_{n}$ as
$A_{n}=D_{\alpha}((U^{n}_{\beta})_{\beta<\alpha})$ with
$U_{\beta}^{n}\in\boldsymbol{\Sigma}^{0}_{1}(X)$. Then
$A=\bigcup_{n\in\omega}(A_{n}\cap
V_{n})=D_{\alpha}((\bigcup_{n\in\omega}(U^{n}_{\beta}\cap
V_{n}))_{\beta<\alpha})$, hence $A\in
D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$. ∎

Notice that if $\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(X)$
and $\boldsymbol{\Gamma}\subseteq\boldsymbol{\Delta}^{0}_{2}(X)$ for a given
zero-dimensional Polish space $X$, then
$\boldsymbol{\Gamma}=D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$ or
$\boldsymbol{\Gamma}=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$
for some $\beta<\omega_{1}$. (Recall that by convention
$D_{0}(\boldsymbol{\Sigma}^{0}_{1}(X))=\\{\emptyset\\}$.) Indeed, if
$\rho\colon\mathcal{N}\to X$ is any retraction then
$\rho^{-1}(\boldsymbol{\Gamma})\subseteq\boldsymbol{\Delta}^{0}_{2}(\mathcal{N})$
is nonselfdual, and thus
$\rho^{-1}(\boldsymbol{\Gamma})=D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(\mathcal{N}))$
or
$\rho^{-1}(\boldsymbol{\Gamma})=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(\mathcal{N}))$
for some $\beta<\omega_{1}$ by Wadge’s analysis of the first $\omega_{1}$-many
nonselfdual pairs in $\mathscr{W}_{\mathcal{N}}$: since
$\boldsymbol{\Gamma}=(\rho^{-1}(\boldsymbol{\Gamma}))(X)$ by Lemma 3.4, we are
done. This reproves [CMM22, Theorem 11.2], and since $\mathscr{P}(X)$ is
selfdual then Proposition 3.14 sharpens it by providing an upper bound on the
possible $\beta$’s when $X$ is countable.

######  Corollary 3.15.

Let $X$ be a countable Polish space and
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(X)$. Then
$\boldsymbol{\Gamma}=D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$ or
$\boldsymbol{\Gamma}=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$
_for some $\beta<||X||_{\operatorname{CB}}$_.

We are now going to show that such bound is optimal.

######  Definition 3.16.

Given $\alpha<\omega_{1}$, set

$A_{\alpha}=\\{x\in
K_{\alpha+1}\mid||x,K_{\alpha+1}||_{\operatorname{CB}}\mbox{ has parity
opposite to }\alpha\\}.$

######  Proposition 3.17.

For all $\alpha<\omega_{1}$, $A_{\alpha}\in
D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(K_{\alpha+1}))\setminus\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(K_{\alpha+1}))$.

###### Proof.

By definition, $A_{0}=\emptyset$ and $K_{\alpha+1}\setminus
A_{\alpha}=\operatorname{ptgl}(A_{\beta})$ if $\alpha=\beta+1$. Moreover,
since when $\alpha$ is limit the choice of the sequence
$(\alpha_{n})_{n\in\omega}$ in the definition of
$K_{\alpha+1}=\operatorname{ptgl}((K_{\alpha_{n}+1})_{n\in\omega})$ is
irrelevant (up to homeomorphism), we can assume that the ordinals $\alpha_{n}$
are all odd, so that also in the limit case $K_{\alpha+1}\setminus
A_{\alpha}=\operatorname{ptgl}((A_{\alpha_{n}})_{n\in\omega})$.

Recall the Wadge degrees $\mathbf{D}_{\alpha}\in\mathscr{W}_{\mathcal{N}}$
from (2.1). Since
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1})\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$,
it is enough to prove by induction that
$A_{\alpha}\equiv_{\mathsf{W}}^{K_{\alpha+1},\mathcal{N}}B_{\alpha}$ for
some/any $B_{\alpha}\in\mathbf{D}_{\alpha}$. If $\alpha=0$ the result is
obvious because $A_{0}=\emptyset=B_{0}$. Suppose now that $\alpha=\beta+1$ and
fix $B_{\beta}\in\mathbf{D}_{\beta}$. Let $f\colon K_{\beta+1}\to\mathcal{N}$
and $g\colon\mathcal{N}\to K_{\beta+1}$ witness
$A_{\beta}\leq_{\mathsf{W}}^{K_{\beta+1},\mathcal{N}}B_{\beta}$ and
$B_{\beta}\leq_{\mathsf{W}}^{\mathcal{N},K_{\beta+1}}A_{\beta}$, respectively.
Let $B_{\alpha}\in\mathbf{D}_{\alpha}$ be defined as in (2.1), that is
$B_{\alpha}=\\{(0)^{n}{{}^{\smallfrown}}(m+1){{}^{\smallfrown}}x\mid
n,m\in\omega\wedge x\in\neg B_{\beta}\\}$. Using $K_{\alpha+1}\setminus
A_{\alpha}=\operatorname{ptgl}(A_{\beta})$ it is straightforward to see that
$\operatorname{ptgl}(f)\colon
K_{\alpha+1}\to\operatorname{ptgl}(\mathcal{N})\subseteq\mathcal{N}$ witnesses
$A_{\alpha}\leq_{\mathsf{W}}^{K_{\alpha+1},\mathcal{N}}B_{\alpha}$, and that
the map $\operatorname{ptgl}(g)\circ\rho\colon\mathcal{N}\to K_{\alpha+1}$,
where $\rho\colon\mathcal{N}\to\operatorname{ptgl}(\mathcal{N})$ is the
retraction defined by $\rho((0)^{\infty})=(0)^{\infty}$ and
$\rho((0)^{n}{{}^{\smallfrown}}(m+1){{}^{\smallfrown}}x)=(0)^{n}{{}^{\smallfrown}}(1){{}^{\smallfrown}}x$,
witnesses $B_{\alpha}\leq_{\mathsf{W}}^{\mathcal{N},K_{\alpha+1}}A_{\alpha}$.
The limit case is similar. ∎

As a by-product, we obtain a computation of $||X||_{\operatorname{CB}}$ for
countable spaces $X$ in terms of difference classes.

######  Proposition 3.18.

Let $X$ be a countable Polish space. Then

$||X||_{\operatorname{CB}}=\min\\{\alpha<\omega_{1}\mid
D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))\\}.$

###### Proof.

By Proposition 3.14 we know that
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))=\mathscr{P}(X)=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$
for $\alpha=||X||_{\operatorname{CB}}$, so it is enough to show that
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))\neq\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$
for every $\alpha<||X||_{\operatorname{CB}}$.

Since $\alpha+1\leq||X||_{\operatorname{CB}}$, by Proposition 3.12 we can
suppose that $K_{\alpha+1}$ is, up to homeomorphism, a closed subset of $X$.
By Proposition 3.17 we get $A_{\alpha}\in
D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(K_{\alpha+1}))\setminus\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(K_{\alpha+1}))$.
Fix a retraction $\rho\colon X\to K_{\alpha+1}$ and set
$A^{\prime}_{\alpha}=\rho^{-1}(A_{\alpha})$: since
$A_{\alpha}\equiv_{\mathsf{W}}^{K_{\alpha+1},X}A^{\prime}_{\alpha}$, we get
$A^{\prime}_{\alpha}\in
D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))\setminus\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$,
as desired. ∎

Combining Corollary 3.15 and Proposition 3.18 we get a full description of
nonselfdual Wadge classes in countable Polish spaces which nicely complements
Theorem 2.9(2) and Lemma 3.13. Given any $\alpha<\Theta$, we set
$\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})\restriction\alpha=\\{\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})\mid||\boldsymbol{\Gamma}||_{\mathcal{N}}<\alpha\\}$.

######  Corollary 3.19.

Let $X$ be a countable Polish space. Then the nonselfdual Wadge pointclasses
in $X$ are precisely $D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$ and their
duals, for $\beta<||X||_{\operatorname{CB}}$.

In particular,
$\mathrm{Class}_{\operatorname{NSD}}(X)=\\{\boldsymbol{\Gamma}(X)\mid\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})\restriction
2\cdot||X||_{\operatorname{CB}}\\}$, and indeed
$\langle\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})\restriction
2\cdot||X||_{\operatorname{CB}},{\subseteq}\rangle$ and
$\langle\mathrm{Class}_{\operatorname{NSD}}(X),{\subseteq}\rangle$ are
isomorphic via the map $\boldsymbol{\Gamma}\mapsto\boldsymbol{\Gamma}(X)$.

##  4\. Proof of 1

In view of the known facts from Propositions 2.2, 2.3, and 2.5, to completely
describe $\mathscr{W}_{X}$ (up to isomorphism) for a given zero-dimensional
Polish space $X$ we need to

  1. (1)

determine which $3\leq\alpha<\Theta_{X}$ (if $\Theta_{X}>3$) are selfdual and
which are not, namely, determine the sets $\mathrm{Ord}_{\mathrm{SD}}(X)$ and
$\mathrm{Ord}_{\mathrm{NSD}}(X)$ from Definition 2.4;

  2. (2)

determine the value of $\Theta_{X}$ when $X$ is countable.

The first of these two goals splits into cases, depending on whether $\alpha$
is successor or not. For successor ordinals (Section 4.1), we will show that,
as in the case of $\mathcal{N}$ and $\mathcal{C}$, nonselfdual pairs and
selfdual degrees alternate. This is relatively easy for uncountable spaces,
although the proof necessarily differs from the one used in the case of
$\mathcal{N}$ and $\mathcal{C}$ because we can no longer rely on the
combinatorics of the space at hand, while in the countable case we need to
perform a deeper analysis of the whole $\mathscr{W}_{X}$ and in particular of
the maximal class(es).

For $\alpha$ limit (Section 4.2), we will immediately observe that for all
$X$, if $\alpha$ has uncountable cofinality then it is nonselfdual. The
situation for limit ordinals of countable cofinality is more delicate: they
can be selfdual, as it happens in $\mathcal{N}$, or not, as in $\mathcal{C}$.
It turns out that for a general zero-dimensional Polish space $X$ the two
behaviours (Baire-like and Cantor-like) can coexist, but in an orderly manner:
if we denote by $\mathsf{Cof}_{\omega}$ the set of limit ordinals of countable
cofinality, then $\mathrm{Ord}_{\mathrm{SD}}(X)\cap\mathsf{Cof}_{\omega}$ is
always an initial segment of $\Theta_{X}\cap\mathsf{Cof}_{\omega}$, which
might be empty, have length $\Theta_{X}$ (i.e.
$\mathrm{Ord}_{\mathrm{NSD}}(X)\cap\mathsf{Cof}_{\omega}=\emptyset$), or might
be of the form $\alpha_{X}\cap\mathsf{Cof}_{\omega}$ for some countable
ordinal $\alpha_{X}>\omega$.

Finally, the deeper analysis of the (non)selfdual levels when $X$ is countable
will also allow us to compute the value of $\Theta_{X}$ (Section 4.3), thus
completing the proof of 1.

###  4.1. The alternating property

We say that a zero-dimensional Polish space $X$ satisfies the _alternating
property_ if for all $\alpha\geq 1$ such that $\alpha+1<\Theta_{X}$ we have

$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)\iff\alpha+1\in\mathrm{Ord}_{\mathrm{NSD}}(X).$

The goal of this subsection is to prove that all zero-dimensional Polish
spaces satisfy the alternating property. The following proposition shows that
there cannot be two consecutive selfdual Wadge classes. In the case of
$\mathcal{N}$ and $\mathcal{C}$ this is done constructively: there are
explicit procedures to construct, starting from a given selfdual set, a
nonselfdual pair immediately after it in the Wadge quasi-order. In the general
case, instead, we cannot rely on any specific combinatorial property of the
space at hand, thus we have to use a different and non-constructive proof.

######  Proposition 4.1 (${\sf AD}$).

Let $X$ be a zero-dimensional Polish space, and
$A,A^{\prime}\in\mathrm{Set}_{\operatorname{SD}}(X)$ be such that
$A<_{\mathsf{W}}^{X}A^{\prime}$. Then there is
$A^{\prime\prime}\in\mathrm{Set}_{\operatorname{NSD}}(X)$ satisfying
$A<_{\mathsf{W}}^{X}A^{\prime\prime}<_{\mathsf{W}}^{X}A^{\prime}$.

###### Proof.

Apply Proposition 2.6 to $A^{\prime}$ to get a clopen partition
$(V_{n})_{n\in\omega}$ of $X$ and nonselfdual sets $(A_{n})_{n\in\omega}$ such
that $A_{n}<_{\mathsf{W}}^{X}A^{\prime}$ and
$A^{\prime}=\bigcup_{n\in\omega}(A_{n}\cap V_{n})$. If
$A_{n}\leq_{\mathsf{W}}^{X}A$ for all $n\in\omega$ via some continuous map
$f_{n}\colon X\to X$, then $f=\bigcup_{n\in\omega}(f_{n}\restriction V_{n})$
would witness $A^{\prime}\leq_{\mathsf{W}}^{X}A$, a contradiction. Since $A$
is selfdual, this means that by Wadge’s Lemma there is $\bar{n}\in\omega$ such
that $A<_{\mathsf{W}}^{X}A_{\bar{n}}$. Setting $A^{\prime\prime}=A_{\bar{n}}$
we are done. ∎

In particular, Proposition 4.1 already yields one implication of the
alternating property.

######  Corollary 4.2 (${\sf AD}$).

Let $X$ be a zero-dimensional Polish space and $\alpha\geq 1$ be such that
$\alpha+1<\Theta_{X}$. If $\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$, then
$\alpha+1\in\mathrm{Ord}_{\mathrm{NSD}}(X)$.

The following corollary will be used in Proposition 4.5. To simplify the
notation let
$D_{\beta}^{*}(\boldsymbol{\Sigma}^{0}_{1}(X))=(D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X)))^{*}=D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))\cup\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$
be the coarse Wadge class associated to
$D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$.

######  Corollary 4.3.

Let $X$ be a countable111111Since we are dealing with a countable space, Borel
determinacy suffices to apply our previous results without appealing to the
full ${\sf AD}$. Polish space. Then all sets $A\subseteq X$ such that
$A\notin\bigcup_{\beta<||X||_{\operatorname{CB}}}D_{\beta}^{*}(\boldsymbol{\Sigma}^{0}_{1}(X))$,
if there exist any, are selfdual, Wadge equivalent to each other, and
$\mathscr{P}(X)=A{\downarrow}_{X}$.

###### Proof.

By Corollary 3.19, all sets $A$ as in the statement are selfdual. Assume
towards a contradiction that there is
$A^{\prime}\notin\bigcup_{\beta<||X||_{\operatorname{CB}}}D_{\beta}^{*}(\boldsymbol{\Sigma}^{0}_{1}(X))$
such that $A^{\prime}\not\leq_{\mathsf{W}}^{X}A$, so that
$A^{\prime}\in\mathrm{Set}_{\operatorname{SD}}(X)$ and
$A<_{\mathsf{W}}^{X}A^{\prime}$. If $A^{\prime}$ is
$\leq_{\mathsf{W}}^{X}$-minimal with such properties, then we get a
contradiction with (the restriction to Borel sets of) Proposition 4.1. Thus
$A\equiv_{\mathsf{W}}^{X}A^{\prime}$ for all
$A,A^{\prime}\notin\bigcup_{\beta<||X||_{\operatorname{CB}}}D_{\beta}^{*}(\boldsymbol{\Sigma}^{0}_{1}(X))$.

Finally, fix $A$ as in the statement and consider an arbitrary $B\subseteq X$.
If $B\in D_{\beta}^{*}(\boldsymbol{\Sigma}^{0}_{1}(X))$ for some
$\beta<||X||_{\operatorname{CB}}$, then $B<_{\mathsf{W}}^{X}A$ by Wadge’s
Lemma; otherwise, $B\equiv_{\mathsf{W}}^{X}A$ by the previous paragraph. In
all cases $B\leq_{\mathsf{W}}^{X}A$, thus $\mathscr{P}(X)=A{\downarrow}_{X}$.
∎

######  Theorem 4.4 (${\sf AD}$).

All uncountable zero-dimensional Polish spaces satisfy the alternating
property.

###### Proof.

Let $X$ be an uncountable Polish zero-dimensional space, and consider any
$1\leq\alpha<\Theta_{X}=\Theta$. By Corollary 4.2 we only need to show that if
$\alpha\in\mathrm{Ord}_{\mathrm{NSD}}(X)$, then
$\alpha+1\in\mathrm{Ord}_{\mathrm{SD}}(X)$. By Theorem 2.9(1) there is
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$ such
that $\boldsymbol{\Gamma}(X)\in\mathrm{Class}_{\operatorname{NSD}}(X)$ and
$||\boldsymbol{\Gamma}(X)||=\alpha$. Since $X$ is uncountable then
$\operatorname{ker}_{\operatorname{CB}}(X)$ is uncountable as well, thus we
can partition $X$ into two uncountable clopen sets $X_{0},X_{1}$. By Theorem
2.9(2) we have
$\boldsymbol{\Gamma}(X_{0})\in\mathrm{Class}_{\operatorname{NSD}}(X_{0})$ and
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}(X_{1})\in\mathrm{Class}_{\operatorname{NSD}}(X_{1})$.
By Lemma 3.6 the smallest Wadge class containing
$\boldsymbol{\Gamma}(X)\cup\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\boldsymbol{\Gamma}}-\widthof{\boldsymbol{\Gamma}}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\boldsymbol{\Gamma}}{\tmpbox}(X)$
is selfdual, i.e. $\alpha+1\in\mathrm{Ord}_{\mathrm{SD}}(X)$. ∎

Combined with Theorem 2.9 and Lemma 3.4, the alternating property implies that
the embedding from $\mathscr{W}_{X}$ into $\mathscr{W}_{\mathcal{N}}$ induced
by any retraction $\rho\colon\mathcal{N}\to X$ (Section 2.2) is almost
surjective and might miss only limit levels of $\mathscr{W}_{X}$. In
particular, except for the countable cofinality case the rank of
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\mathsf{W}}(\mathcal{N})$ and the rank
(in $X$) of $\boldsymbol{\Gamma}(X)$ are at distance at most $1$.

We now move to countable spaces.

######  Proposition 4.5.

Let $X$ be a countable Polish space. Then $\mathscr{P}(X)$ is a Wadge class if
and only if $X$ is not simple.

###### Proof.

Suppose first that $X$ is simple, so that in particular
$||X||_{\operatorname{CB}}=\alpha+1$ for some $\alpha<\omega_{1}$. By
Corollary 3.19 we know that
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))\neq\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$,
while the “moreover” part of Proposition 3.14 gives
$\mathscr{P}(X)=D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))\cup\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$.
It follows that we cannot have $\mathscr{P}(X)=A{\downarrow}_{X}$ for
$A\subseteq X$, because if $B\subseteq X$ is complete for
$D_{\alpha}(\boldsymbol{\Sigma}^{0}_{1}(X))$ then $B,\neg
B\leq_{\mathsf{W}}^{X}A$ would imply $B<_{\mathsf{W}}^{X}A$ by
$B\not\leq_{\mathsf{W}}^{X}\neg B$, a contradiction.

Suppose now that $X$ is not simple. By Corollary 4.3 it is enough to show that
there is some $A\subseteq X$ such that
$A\notin\bigcup_{\beta<||X||_{\operatorname{CB}}}D_{\beta}^{*}(\boldsymbol{\Sigma}^{0}_{1}(X))$.
We distinguish two cases.

If $||X||_{\operatorname{CB}}=\lambda$ is limit, fix a strictly increasing
sequence $(\alpha_{i})_{i\in\omega}$ cofinal in $\lambda$. By Lemma 2.10 we
can find a clopen partition $(C_{n})_{n\in\omega}$ of $X$ such that
$||C_{n}||_{\operatorname{CB}}<\lambda$ and
$\sup_{n\in\omega}||C_{n}||_{\operatorname{CB}}=\lambda$. Fix an injection
$i\mapsto n_{i}$ such that $||C_{n_{i}}||_{\operatorname{CB}}>\alpha_{i}$, so
that
$D_{\alpha_{i}}(\boldsymbol{\Sigma}^{0}_{1}(C_{n_{i}}))\in\mathrm{Class}_{\operatorname{NSD}}(C_{n_{i}})$
for all $i\in\omega$ by Corollary 3.19. By Lemma 3.6 there is a selfdual Wadge
class $A{\downarrow}_{X}$ which contains
$\bigcup_{i\in\omega}D_{\alpha_{i}}(\boldsymbol{\Sigma}^{0}_{1}(X))$, hence
$A\notin\bigcup_{\beta<\lambda}D_{\beta}^{*}(\boldsymbol{\Sigma}^{0}_{1}(X))$
because all the pointclasses $D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$ are
distinct and nonselfdual by Corollary 3.19.

Suppose now that $\operatorname{tp}(X)=(\beta+1,N)$ with $1<N\leq\omega$. By
Lemma 2.10 again we can find a clopen partition $(C_{n})_{n<N}$ of $X$ such
that $||C_{n}||_{\operatorname{CB}}=\beta+1$, so that
$\boldsymbol{\Gamma}_{0}(C_{0})=D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(C_{0}))\in\mathrm{Class}_{\operatorname{NSD}}(C_{0})$
and
$\boldsymbol{\Gamma}_{1}(C_{1})=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(C_{1}))\in\mathrm{Class}_{\operatorname{NSD}}(C_{1})$
by Corollary 3.19. (For the remaining $1<n<N$ the choice of the nonselfdual
pointclasses $\boldsymbol{\Gamma}_{n}(C_{n})$ is irrelevant.) Then Lemma 3.6
allows us to conclude that there is a selfdual $A\subseteq X$ with
$A{\downarrow}_{X}\supseteq
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))\cup\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$,
which is again as required by Corollary 3.19. ∎

######  Theorem 4.6.

All countable zero-dimensional Polish spaces satisfy the alternating property.

###### Proof.

Let $X$ be a countable Polish space. By Corollary 4.2, we only need to prove
that if $\alpha\in\mathrm{Ord}_{\mathrm{NSD}}(X)$ and $\alpha+1<\Theta_{X}$
then $\alpha+1\in\mathrm{Ord}_{\mathrm{SD}}(X)$. By Corollary 3.19 there
exists a $\beta<||X||_{\operatorname{CB}}$ such that
$||D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))||=\alpha$. We distinguish two
cases.

Assume first that $\beta+1<||X||_{\operatorname{CB}}$. Then
$D^{\beta+1}_{\operatorname{CB}}(X)\neq\emptyset$ and
$D^{\beta}_{\operatorname{CB}}(X)$ is infinite, which implies that we can
split $X$ in two clopen sets $X_{0}$ and $X_{1}$ with
$||X_{i}||_{\operatorname{CB}}\geq\beta+1$ for $i=0,1$. (For example, pick
$x\in D^{\beta}_{\operatorname{CB}}(X)\setminus
D^{\beta+1}_{\operatorname{CB}}(X)$ and $X_{0}$ clopen such that $X_{0}\cap
D^{\beta}_{\operatorname{CB}}(X)=\\{x\\}$: then $X_{0}$ and $X_{1}=X\setminus
X_{0}$ are as desired.) By Corollary 3.19 we have that
$\boldsymbol{\Gamma}_{0}(X_{0})=D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X_{0}))\in\mathrm{Class}_{\operatorname{NSD}}(X_{0})$
and
$\boldsymbol{\Gamma}_{1}(X_{1})=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{D}-\widthof{D}/4]{\kern-0.3pt\rotatebox[origin={c}]{180.0}{$\bigwedge$}\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{D}{\tmpbox}_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X_{1}))\in\mathrm{Class}_{\operatorname{NSD}}(X_{1})$,
so by Lemma 3.6 their supremum is selfdual and
$\alpha+1\in\mathrm{Ord}_{\mathrm{SD}}(X)$.

Assume now $||X||_{\operatorname{CB}}=\beta+1$. If $X$ were simple, then
$\Theta_{X}=||D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))||+1=\alpha+1$ by
Proposition 3.14, contradicting the choice of $\alpha$. Thus $X$ is not simple
and Proposition 4.5 together with Corollary 4.3 show that at level $\alpha+1$
we have the selfdual Wadge class $\mathscr{P}(X)$, so
$\alpha+1\in\mathrm{Ord}_{\mathrm{SD}}(X)$ again and we are done. ∎

###  4.2. Limit levels

We now move to the analysis of limit levels of $\mathscr{W}_{X}$. We first
settle the uncountable cofinality case.

######  Proposition 4.7 (${\sf AD}$).

Let $X$ be a zero-dimensional Polish space. If $1\leq\alpha<\Theta_{X}$ is a
limit ordinal and $\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$, then
$\operatorname{cof}(\alpha)=\omega$. Thus at limit levels of uncountable
cofinality there is always a nonselfdual pair.

###### Proof.

Let $A\in\mathrm{Set}_{\operatorname{SD}}(X)$ be such that
$||A||_{\mathsf{W}}^{X}=\alpha$. By Proposition 2.6, there are sets
$(A_{n})_{n\in\omega}$ such that
$\alpha=\sup_{n\in\omega}(||A_{n}||_{\mathsf{W}}^{X}+1)$: this already shows
that $\alpha$ has countable cofinality. ∎

To deal with the countable cofinality case, we need to distinguish whether
$\operatorname{ker}_{\operatorname{CB}}(X)$ is compact or not. We begin with a
technical lemma, which might be of independent interest because it shows that
$\omega\times\mathcal{C}$ is minimal for closed embeddability among zero-
dimensional Polish spaces with non-compact perfect kernel.

######  Lemma 4.8.

Let $X$ be a zero-dimensional Polish space. The following are equivalent:

  1. (i)

$\operatorname{ker}_{\operatorname{CB}}(X)$ is not compact;

  2. (ii)

there exists an infinite (countable) clopen partition $(C_{n})_{n\in\omega}$
of $X$ such that each $C_{n}$ is uncountable;

  3. (iii)

$X$ contains a closed set $F$ homeomorphic to $\omega\times\mathcal{C}$.

###### Proof.

If $X$ is countable, then all of (i)–(iii) are false, so without loss of
generality we may assume that $X$ is uncountable.

We start with (i) $\Rightarrow$ (ii). Since
$\operatorname{ker}_{\operatorname{CB}}(X)$ is not compact, by 3.8 there is an
infinite clopen partition $(C^{\prime}_{n})_{n\in\omega}$ of
$\operatorname{ker}_{\operatorname{CB}}(Z)$, and since
$\operatorname{ker}_{\operatorname{CB}}(X)$ is perfect each $C^{\prime}_{n}$
is uncountable. Thus it is enough to set $C_{n}=\rho^{-1}(C^{\prime}_{n})$
where $\rho\colon X\to\operatorname{ker}_{\operatorname{CB}}(X)$ is any
retraction.

Assume now that (ii) holds. Since the $C_{n}$’s are uncountable Polish spaces,
they contain a closed set $F_{n}\subseteq C_{n}$ homeomorphic to
$\mathcal{C}$: then $F=\bigcup_{n\in\omega}F_{n}$ is still closed and
homeomorphic to $\omega\times\mathcal{C}$, hence (iii) holds.

Finally, we prove (iii) $\Rightarrow$ (i). Since $F$ is perfect, then
$F\subseteq\operatorname{ker}_{\operatorname{CB}}(X)$. But then
$\operatorname{ker}_{\operatorname{CB}}(X)$ cannot be compact, otherwise so
would be $F$ and $\omega\times\mathcal{C}$, which is false. ∎

######  Theorem 4.9 (${\sf AD}$).

Let $X$ be an uncountable zero-dimensional Polish space, so that
$\Theta_{X}=\Theta$. Then the following are equivalent:

  1. (i)

$\operatorname{ker}_{\operatorname{CB}}(X)$ is not compact;

  2. (ii)

$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$ for every limit $\alpha<\omega_{1}$;

  3. (iii)

if $\alpha\in\Theta_{X}\cap\mathsf{Cof}_{\omega}$, then
$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$.

###### Proof.

It is enough to prove (i) $\Rightarrow$ (iii) and (the contrapositive of) (ii)
$\Rightarrow$ (i).

Assume first that $\operatorname{ker}_{\operatorname{CB}}(X)$ is not compact,
so that by Lemma 4.8 there is an infinite clopen partition
$(C_{n})_{n\in\omega}$ of $X$ into uncountable pieces. Given any
$\alpha\in\Theta_{X}\in\mathsf{Cof}_{\omega}$, pick a strictly increasing
sequence of ordinals $(\alpha_{n})_{n\in\omega}$ cofinal in $\alpha$ such that
$\alpha_{n}\in\mathrm{Ord}_{\mathrm{NSD}}(X)$ for all $n\in\omega$ (which
exists by the alternating property), and using Theorem 2.9(1) let
$\boldsymbol{\Gamma}_{n}\in\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$
be such that $||\boldsymbol{\Gamma}_{n}(X)||_{X}=\alpha_{n}$. By Theorem
2.9(2) and uncountability of $C_{n}$, we have
$\boldsymbol{\Gamma}_{n}(C_{n})\in\mathrm{Class}_{\operatorname{NSD}}(C_{n})$
as well, hence by Lemma 3.6 the supremum
$\boldsymbol{\Gamma}\in\mathrm{Class}_{\mathsf{W}}(X)$ of the pointlcasses
$\boldsymbol{\Gamma}_{n}(X)$ is selfdual, i.e.
$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$.

Assume now that $\operatorname{ker}_{\operatorname{CB}}(X)$ is compact and let
$\beta=||X||_{\operatorname{CB}}$. Consider any
$\beta+\omega<\alpha<\omega_{1}$: we want to show that if
$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$ then $\alpha$ is a successor ordinal,
so that any limit ordinal between $\beta+\omega$ and $\omega_{1}$ witnesses
the failure of (ii). Let $A\subseteq X$ be a selfdual set such that
$||A||_{\mathsf{W}}^{X}=\alpha$. By Proposition 2.6 there exists a clopen
partition $(V_{n})_{n\in\omega}$ of $X$ and sets $(A_{n})_{n\in\omega}$ such
that $A_{n}<_{\mathsf{W}}^{X}A=\bigcup_{n\in\omega}(A_{n}\cap V_{n})$ and
$\alpha=\sup_{n\in\omega}(\alpha_{n}+1)$ for
$\alpha_{n}=||A_{n}||_{\mathsf{W}}^{X}$. Recall also from Remark 2.7(1) that
we can assume that for each $n\in\omega$ either $A_{n}=X$, or else
$A_{n}\subseteq V_{n}$. Since $\operatorname{ker}_{\operatorname{CB}}(X)$ is
compact, there is $N\in\omega$ such that
$V_{n}\cap\operatorname{ker}_{\operatorname{CB}}(X)=\emptyset$ for all $n\geq
N$. Fix any $n\geq N$. Then $||V_{n}||_{\operatorname{CB}}\leq\beta$ by Lemma
2.11, hence if $A_{n}\subseteq V_{n}$ then $A_{n}\in
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(V_{n}))\subseteq
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$ by Proposition 3.14, which means
that $\alpha_{n}<\beta+\omega$ because by Theorem 2.9 and the alternating
property
$||D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))||_{X}\leq||D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(\mathcal{N}))||_{\mathcal{N}}=2\beta+1$.
If instead $A_{n}=X$, then $\alpha_{n}=1$. Hence $\alpha_{n}+1<\beta+\omega$
for all $n\geq N$. Since $\alpha>\beta+\omega$, this means that necessarily
$\alpha=\sup_{n\in\omega}(\alpha_{n}+1)=\sup_{n<N}(\alpha_{n}+1)=\max_{n<N}(\alpha_{n}+1)$,
and thus $\alpha$ is a successor ordinal. ∎

If $X$ is countable, so that
$\operatorname{ker}_{\operatorname{CB}}(X)=\emptyset$ is trivially compact,
then $\Theta_{X}<\omega_{1}<\Theta=\Theta_{\mathcal{N}}$ (see Section 4.3 for
the exact computation of $\Theta_{X}$), and thus $\mathscr{W}_{X}$ cannot be
isomorphic to $\mathscr{W}_{\mathcal{N}}$. Combining this with Theorem 4.9 and
our previous results (and recalling that, up to homeomorphism, any zero-
dimensional Polish space can be construed as a closed subspace of
$\mathcal{N}$) we get:

######  Corollary 4.10 (${\sf AD}$).

If $X$ is a zero-dimensional Polish space, then $\mathscr{W}_{X}$ is order-
isomorphic to $\mathscr{W}_{\mathcal{N}}$ if and only if
$\operatorname{ker}_{\operatorname{CB}}(X)$ is not compact, where (when it
exists) the isomorphism is canonically121212Since we are working in models of
${\sf AD}$, where the Axiom of Choice ${\sf AC}$ fails, showing that two
partial orders have the same description is not enough to conclude that they
are actually isomorphic, so it is very important to observe that in our case
the isomorphism can be realized in a very canonical and definable way. induced
by $A\mapsto\rho^{-1}(A)$ for some/any retraction $\rho\colon\mathcal{N}\to
X$.

We now move to spaces $X$ with a compact perfect kernel (this includes the
case of countable spaces). It turns out that the compact rank from Section 3.3
provides a crucial dividing line.

######  Theorem 4.11 (${\sf AD}$).

Let $X$ be a zero-dimensional Polish space such that
$\operatorname{ker}_{\operatorname{CB}}(X)$ is compact, and let
$\alpha\in\Theta_{X}\cap\mathsf{Cof}_{\omega}$.

  1. (1)

If $\alpha>||X||_{\operatorname{Comp}}$, then
$\alpha\in\mathrm{Ord}_{\mathrm{NSD}}(X)$.

  2. (2)

If $\alpha<||X||_{\operatorname{Comp}}$, then
$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$.

  3. (3)

If $\alpha=||X||_{\operatorname{Comp}}$, then
$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$ if and only if there is a clopen
partition $(V_{n})_{n\in\omega}$ of $X$ and $N\in\omega$ such that
$\alpha=\sup_{n\geq N}(||V_{n}||_{\operatorname{CB}}+1)$.

###### Proof.

(1) Suppose towards a contradiction that $\alpha>||X||_{\operatorname{Comp}}$
but $\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$: we adapt the proof of Theorem
4.9 to show that $\alpha$ needs to be a successor ordinal, contradicting
$\alpha\in\mathsf{Cof}_{\omega}$. Given any
$A\in\mathrm{Set}_{\operatorname{SD}}(X)$ such that
$||A||_{\mathsf{W}}^{X}=\alpha$, use Proposition 2.6 to get a clopen partition
$(V_{n})_{n\in\omega}$ of $X$ and sets $(A_{n})_{n\in\omega}$ such that
$A_{n}<_{\mathsf{W}}^{X}A=\bigcup_{n\in\omega}(A_{n}\cap V_{n})$ and
$\alpha=\sup_{n\in\omega}(\alpha_{n}+1)$ for
$\alpha_{n}=||A_{n}||_{\mathsf{W}}^{X}$. Since by definition
$D^{||X||_{\operatorname{Comp}}}_{\operatorname{CB}}(X)$ is compact, there
exists $N\in\omega$ such that $V_{n}\subseteq X\setminus
D^{||X||_{\operatorname{Comp}}}_{\operatorname{CB}}(X)$ for all $n\geq N$.
Using that either $A_{n}=X$ or $A_{n}\subseteq V_{n}$, this implies that
$A_{n}\in D_{||X||_{\operatorname{Comp}}}(\boldsymbol{\Sigma}^{0}_{1}(X))$ for
all $n\geq N$. Arguing as in Theorem 4.9 (and using that
$\alpha>||X||_{\operatorname{Comp}}$ is limit), we get that
$||D_{||X||_{\operatorname{Comp}}}(\boldsymbol{\Sigma}^{0}_{1}(X))||_{X}\leq||D_{||X||_{\operatorname{Comp}}}(\boldsymbol{\Sigma}^{0}_{1}(\mathcal{N}))||_{\mathcal{N}}=2\cdot||X||_{\operatorname{Comp}}+1<\alpha$,
hence $\sup_{n\geq N}(\alpha_{n}+1)<\alpha$. Thus
$\alpha=\sup_{n\in\omega}(\alpha_{n}+1)=\max_{n<N}(\alpha_{n}+1)$ is a
successor ordinal, a contradiction.

(2) Suppose now that $\alpha<||X||_{\operatorname{Comp}}$, so that in
particular $X$ is not compact. Then by Lemma 3.10 there is a clopen partition
$(C_{n})_{n}$ of $X$ such that $D^{\alpha+1}_{\operatorname{CB}}(X)\subseteq
C_{0}$ and $||C_{n+1}||_{\operatorname{CB}}=\alpha+1$ for all $n\in\omega$.
For every $n\geq 1$ and $\beta<\alpha$, Corollary 3.19 implies that
$D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(C_{n}))\in\mathrm{Class}_{\operatorname{NSD}}(C_{n})$.
Moreover, $\beta\leq||D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))||\leq
2\beta+1$: if $X$ is countable this follows from the alternating property and
Corollary 3.19 together with
$||X||_{\operatorname{CB}}\geq||X||_{\operatorname{Comp}}>\alpha>\beta$, while
if $X$ is uncountable we can use the alternating property and Lemma 3.13.
Since $\alpha$ is limit, then $2\beta+1<\alpha$ and hence
$\beta\leq||D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))||<\alpha$. Therefore
taking any strictly increasing sequence $(\alpha_{n})_{n\geq 1}$ cofinal in
$\alpha$ we have
$D_{\alpha_{n}}(\boldsymbol{\Sigma}^{0}_{1}(C_{n}))\in\mathrm{Class}_{\operatorname{NSD}}(C_{n})$,
$||D_{\alpha_{n}}(\boldsymbol{\Sigma}^{0}_{1}(X))||_{X}<\alpha$, and
$\sup_{n\geq 1}||D_{\alpha_{n}}(\boldsymbol{\Sigma}^{0}_{1}(X))||=\alpha$.
Setting $\boldsymbol{\Gamma}_{0}=\\{\mathcal{N}\\}$ and
$\boldsymbol{\Gamma}_{n}=D_{\alpha_{n}}(\boldsymbol{\Sigma}^{0}_{1})$ for
$n\geq 1$, and applying Lemma 3.6, we then get
$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$, as desired.

(3) The proof of the desired equivalence is a variation of the arguments for
parts (1) and (2). Assume first that $\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$.
Given any $A\in\mathrm{Set}_{\operatorname{SD}}(X)$ such that
$||A||_{\mathsf{W}}^{X}=\alpha$, let $(V_{n})_{n\in\omega}$ and
$(A_{n})_{n\in\omega}$ be as in Proposition 2.6. Since
$D^{\alpha}_{\operatorname{CB}}(X)$ is compact, there is $N\in\omega$ such
that $V_{n}\cap D^{\alpha}_{\operatorname{CB}}(X)=\emptyset$ for all $n\geq
N$, which also means $||V_{n}||_{\operatorname{CB}}\leq\alpha$ by Lemma 2.11.
If $||V_{n}||_{\operatorname{CB}}=\alpha$ for some $n\geq N$, then we apply
Lemma 2.10 to all such countable spaces $V_{n}$ and get the desired partition.
(Recall that $\alpha$ is limit.) So we can assume that
$||V_{n}||_{\operatorname{CB}}+1\leq\alpha$ for all $n\geq N$ and we only need
to show that for every $\beta<\alpha$ there is $n\geq N$ with
$||V_{n}||_{\operatorname{CB}}\geq\beta$. Assume not: since we can assume that
either $A_{n}=X$ or $A_{n}\subseteq V_{n}$, by Proposition 3.14 we would have
$A_{n}\in
D_{||V_{n}||_{\operatorname{CB}}}(\boldsymbol{\Sigma}^{0}_{1}(V_{n}))\subseteq
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(V_{n}))\subseteq
D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$ for all $n\geq N$, and since the
usual computation gives $||D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))||_{X}\leq
2\beta+1$ and $2\beta+1<\alpha$ because $\alpha$ is limit, then
$\alpha=\sup_{n\in\omega}(||A_{n}||_{\mathsf{W}}^{X}+1)=\max_{n<N}(||A_{n}||_{\mathsf{W}}^{X}+1)$,
contradicting the fact that $\alpha$ is limit.

Conversely, let $(V_{n})_{n\in\omega}$ and $N\in\omega$ be as in part (3), and
fix a sequence $(\alpha_{k})_{k\in\omega}$ cofinal in $\alpha$. Since $\alpha$
is limit, the assumption on $(V_{n})_{n\in\omega}$ implies that for each
$k\in\omega$ we can find $n_{k}\geq N$ such that
$\alpha_{k}<||V_{n_{k}}||_{\operatorname{CB}}$ and further ensure that the map
$k\mapsto n_{k}$ is injective. By Corollary 3.19 the class
$\boldsymbol{\Gamma}_{n_{k}}(V_{n_{k}})=D_{\alpha_{k}}(\boldsymbol{\Sigma}^{0}_{1}(V_{n_{k}}))$
is nonselfdual in $V_{n_{k}}$, and as in the proof of part (2) one can show
that $\alpha_{k}\leq||D_{\alpha_{k}}(\boldsymbol{\Sigma}^{0}_{1}(X))||\leq
2\alpha_{k}+1<\alpha$. Setting $\boldsymbol{\Gamma}_{n}=\\{\mathcal{N}\\}$ for
all $n\in\omega$ which are not of the form $n_{k}$ and applying Lemma 3.6 we
get $\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$, as desired. ∎

######  Remark 4.12.

Together with Lemma 2.10, Theorem 4.11(3) shows that if $X$ is a countable
zero-dimensional Polish space and
$||X||_{\operatorname{CB}}=||X||_{\operatorname{Comp}}=\alpha<\Theta_{X}$ is
limit, then necessarily $\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$.

Concerning the characterization in Theorem 4.11(3), it is easy to provide
examples of both sorts.

######  Example 4.13.

Let $\alpha$ be a countable limit ordinal and fix a strictly increasing
sequence $(\alpha_{n})_{n\in\omega}$ cofinal in $\alpha$. Denote by
$\bigoplus_{n\in\omega}X_{n}$ the (disjoint) sum of the topological spaces
$X_{n}$. (If $X_{n}\subseteq\mathcal{N}$, then up to homeomorphism the sum can
be construed as
$\bigoplus_{n\in\omega}X_{n}=\bigcup_{n\in\omega}\\{n{{}^{\smallfrown}}x\mid
n\in\omega\wedge x\in X_{n}\\}$.) Then the spaces

$Y_{\alpha}=\bigoplus_{n\in\omega}K_{\alpha_{n}+1}\qquad\text{and}\qquad
Z_{\alpha}=\operatorname{ptgl}((\omega\times K_{\alpha_{n}+1})_{n\in\omega})$

are both countable zero-dimensional Polish spaces with compact rank $\alpha$,
and by Theorem 4.11(3) we have
$\alpha\in\mathrm{Ord}_{\mathrm{SD}}(Y_{\alpha})$ but
$\alpha\in\mathrm{Ord}_{\mathrm{NSD}}(Z_{\alpha})$. Moreover,
$||Y_{\alpha}||_{\operatorname{CB}}=\alpha$ and
$||Z_{\alpha}||_{\operatorname{CB}}=\alpha+1$ (indeed,
$\operatorname{tp}(Z_{\alpha})=(\alpha+1,1)$, i.e. $Z_{\alpha}$ is also
simple), and the values for such CB-ranks are as small as possible by Remark
4.12.

The following result, together with Proposition 4.7 and Theorem 4.9, completes
the proof of part (3) of 1. Let $\alpha_{X}$ be the smallest
$\alpha_{X}\leq\Theta_{X}$ such that $\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)$
for all limit $\alpha<\alpha_{X}$ with $\operatorname{cof}(\alpha)=\omega$. By
Theorems 4.9 and 4.11 we get:

######  Corollary 4.14 (${\sf AD}$).

Let $X$ be an arbitrary zero-dimensional Polish space. Then either
$\alpha_{X}=\Theta_{X}=\Theta$ (if $\operatorname{ker}_{\operatorname{CB}}(X)$
is not compact), or else $\alpha_{X}<\omega_{1}$ and
$\alpha\in\mathrm{Ord}_{\mathrm{NSD}}(X)$ for all limit
$\alpha_{X}\leq\alpha<\Theta_{X}$.

Indeed, by Theorems 4.9 and 4.11 the ordinal $\alpha_{X}$ can be computed as
follows:

  * •

If $\operatorname{ker}_{\operatorname{CB}}(X)$ is not compact, then $X$ is
uncountable and $\alpha_{X}=\Theta_{X}=\Theta$.

  * •

If $\operatorname{ker}_{\operatorname{CB}}(X)$ is compact and
$||X||_{\operatorname{Comp}}$ is a successor ordinal, then
$||X||_{\operatorname{Comp}}=\lambda+n$ for some limit or null
$\lambda<\omega_{1}$ and $n\in\omega\setminus\\{0\\}$: set
$\alpha_{X}=\lambda+1$ if $\lambda>0$ and $\alpha_{X}=0$ otherwise.

  * •

If $\operatorname{ker}_{\operatorname{CB}}(X)$ is compact and
$||X||_{\operatorname{Comp}}=\lambda$ is limit, then we have to distinguish
cases depending on whether
$||X||_{\operatorname{Comp}}\in\mathrm{Ord}_{\mathrm{SD}}(X)$ or not. In the
former case, we again set $\alpha_{X}=\lambda+1$. Otherwise we set
$\alpha_{X}=0$ if $\lambda=\omega$, $\alpha_{X}=\lambda^{\prime}+1$ if
$\lambda$ is of the form $\lambda^{\prime}+\omega$ for some limit
$\lambda^{\prime}<\omega_{1}$, and $\alpha_{X}=\lambda$ otherwise (i.e. if
$\lambda$ is a limit of limit ordinals).

In particular, $\alpha_{X}$ is either $0$, or the successor of a countable
limit ordinal, or a limit of countable limit ordinals, or the ordinal
$\Theta$. Moreover, when $\alpha_{X}<\omega_{1}$ we have
$\alpha_{X}\leq||X||_{\operatorname{CB}}$ unless $||X||_{\operatorname{CB}}$
is limit, in which case we might have
$\alpha_{X}\leq||X||_{\operatorname{CB}}$ or
$\alpha_{X}=||X||_{\operatorname{CB}}+1$.

###  4.3. Length of the Wadge hierarchy for countable spaces

By Proposition 2.5 we know that if a zero-dimensional Polish space $X$ is
uncountable, then $\Theta_{X}=\Theta$. Thus to complete the proof of part (4)
of 1 we just need to compute the value of $\Theta_{X}$ when $X$ is countable.
Notice that since we consider only nonempty spaces $X$, the minimal possible
length for $\mathscr{W}_{X}$ is $2$.

######  Theorem 4.15.

Let $X$ be a countable Polish space. Then
$\Theta_{X}=2\cdot||X||_{\operatorname{CB}}+\varepsilon_{X}$, for some
$\varepsilon_{X}\in\\{-1,0,1\\}$.

###### Proof.

Let $||X||_{\operatorname{CB}}=\lambda+n$ for some $\lambda$ limit or null and
$n\in\omega$. We distinguish various cases.

If $\lambda=0$ (i.e. $||X||_{\operatorname{CB}}$ is finite), then $n>0$
because $X\neq\emptyset$. Then by Corollary 3.19 the largest nonselfdual Wadge
classes in $\mathscr{W}_{X}$ are $D_{n-1}(\boldsymbol{\Sigma}^{0}_{1}(X))$ and
its dual, and by the alternating property (and the fact that the Wadge rank
starts by $1$) we have $||D_{n-1}(\boldsymbol{\Sigma}^{0}_{1}(X))||=2n-1$. By
Proposition 3.14 and Proposition 4.5, it follows that
$\Theta_{X}=2n=2\cdot||X||_{\operatorname{CB}}$ if $X$ is simple, in which
case we set $\varepsilon_{X}=0$, and
$\Theta_{X}=2n+1=2\cdot||X||_{\operatorname{CB}}+1$ otherwise, in which case
$\varepsilon_{X}=1$.

Assume now that $\lambda>0$. By the alternating property and Corollary 3.19 we
know that $\Theta_{X}\geq\lambda$ and that the ranks of difference classes of
the form $D_{\beta}(\boldsymbol{\Sigma}^{0}_{1}(X))$ with $\beta<\lambda$ are
cofinal in $\lambda$. So we have two subcases. If $n=0$, then $X$ is not
simple and Proposition 4.5 gives that $\mathscr{P}(X)$ is a selfdual Wadge
class of rank exactly $\lambda$: thus
$\Theta_{X}=\lambda+1=2\cdot||X||_{\operatorname{CB}}+1$ and we set
$\varepsilon_{X}=1$. If instead $n>0$, then by the alternating property and
Corollary 3.19 again we know that there are precisely $n$-many nonselfdual
Wadge classes with rank $\geq\lambda$, i.e. the classes
$D_{\lambda+i}(\boldsymbol{\Sigma}^{0}_{1}(X))$ for $i<n$, which by the
alternating property means that we have at least $2n-1$ Wadge classes with
rank $\geq\lambda$. Then we have to consider two issues. First, if
$\alpha_{X}>\lambda$ then $\lambda\in\mathrm{Ord}_{\mathrm{SD}}(X)$ and
$||D_{\lambda}(\boldsymbol{\Sigma}^{0}_{1}(X))||=\lambda+1$, i.e. we have one
more (selfdual) class before $D_{\lambda}(\boldsymbol{\Sigma}^{0}_{1}(X))$; if
instead $\alpha_{X}\leq\lambda$ then
$||D_{\lambda}(\boldsymbol{\Sigma}^{0}_{1}(X))||=\lambda$ and there is no
additional class to be taken into account. Second, by Proposition 3.14 and
Proposition 4.5 we know that if $X$ is simple then
$D_{\lambda+n-1}(\boldsymbol{\Sigma}^{0}_{1}(X))$ is the last Wadge class in
$\mathscr{W}_{X}$, while if $X$ is not simple then $\mathscr{P}(X)$ forms yet
one more Wadge class immediately after
$D_{\lambda+n-1}(\boldsymbol{\Sigma}^{0}_{1}(X))$ and its dual. Summing up the
discussion above we have that
$\Theta_{X}=2\cdot||X||_{\operatorname{CB}}+\varepsilon_{X}$ where
$\varepsilon_{X}=\beta_{X}+\delta_{X}-1$ and $\beta_{X}=0$ if $X$ is simple
and $\beta_{X}=1$ otherwise, while $\delta_{X}=0$ if $\alpha_{X}\leq\lambda$
and $\delta_{X}=1$ otherwise. ∎

###  4.4. Optimality

The previous results are sharp, meaning that all possible shapes for a Wadge
hierarchy $\mathscr{W}_{X}$ which are coherent with our description are
actually realized by some zero-dimensional Polish space $X$. We actually prove
a finer result, showing that all possible configurations of the parameters
$||X||_{\operatorname{CB}}$, $||X||_{\operatorname{Comp}}$, and, if $X$ is
countable, $\operatorname{tp}(X)$ can be realized in suitable spaces, together
with some additional features relevant to our analysis of $\mathscr{W}_{X}$.
The limitations in the statement follow from the possible values of the CB-
rank and the CB-type (and the fact that we consider only nonempty spaces), the
definition of the compact rank, Lemma 2.10, and Remark 4.12. Notice also that
the additional conditions on $\beta$ and $\gamma$ are relevant only in part
(1): in particular, in part (2) the ordinal $\gamma$ can unconditionally
assume any value $\leq\alpha$.

######  Proposition 4.16.

Let $(\alpha,\beta)\in\omega_{1}\times(\omega+1)$ and $\gamma\leq\alpha$.
Further assume that $\beta\geq 1$, $\beta=\omega$ if $\alpha$ is limit,
$\gamma=\alpha$ if $\beta=\omega$, and $\gamma\leq\alpha^{\prime}$ if
$\alpha=\alpha^{\prime}+1$ and $\beta<\omega$.

  1. (1)

If $\alpha\geq 1$ there is a countable (zero-dimensional) Polish space $X$
with $\operatorname{tp}(X)=(\alpha,\beta)$ and
$||X||_{\operatorname{Comp}}=\gamma$. Moreover, when $\gamma$ is limit we can
require to have $\gamma\in\mathrm{Ord}_{\mathrm{SD}}(X)$, or else to have
$\gamma\in\mathrm{Ord}_{\mathrm{NSD}}(X)$ (provided that $\alpha>\gamma$, in
which case $\alpha$ needs to be a successor ordinal).

  2. (2)

There is an uncountable zero-dimensional Polish space $X$ with
$\operatorname{ker}_{\operatorname{CB}}(X)$ compact,
$||X||_{\operatorname{CB}}=\alpha$, and $||X||_{\operatorname{Comp}}=\gamma$.
Moreover, when $\gamma$ is limit we can freely decide to have
$\gamma\in\mathrm{Ord}_{\mathrm{SD}}(X)$ or
$\gamma\in\mathrm{Ord}_{\mathrm{NSD}}(X)$.

  3. (3)

There is a(n uncountable) zero-dimensional Polish space $X$ with
$\operatorname{ker}_{\operatorname{CB}}(X)$ non-compact and
$||X||_{\operatorname{CB}}=\alpha$.

###### Proof.

(1) Further developing Example 4.13, for any $\delta<\omega_{1}$ we set

$Y_{\delta}=\bigoplus_{n\in\omega}K_{\delta_{n}+1}\qquad\text{and}\qquad
Z_{\delta}=\operatorname{ptgl}((\omega\times K_{\delta_{n}+1})_{n\in\omega}),$

where $(\delta_{n})_{n\in\omega}$ is a sequence cofinal in $\delta$ if the
latter is limit, or the constant sequence with value $\delta^{\prime}$ if
$\delta=\delta^{\prime}+1$. Then
$\operatorname{tp}(Y_{\delta})=(\delta,\omega)$,
$\operatorname{tp}(Z_{\delta})=(\delta+1,1)$, and
$||Y_{\delta}||_{\operatorname{Comp}}=||Z_{\delta}||_{\operatorname{Comp}}=\delta$.
Moreover, if $\delta$ is limit then
$\delta\in\mathrm{Ord}_{\mathrm{SD}}(Y_{\delta})$ while
$\delta\in\mathrm{Ord}_{\mathrm{NSD}}(Z_{\delta})$.

The space $X=Y_{\alpha}$ already settles the case when $\alpha$ is limit or
when $\alpha$ is a successor but $\beta=\omega$. So assume that
$\alpha=\alpha^{\prime}+1$ and $1\leq\beta<\omega$, in which case
$\gamma\leq\alpha^{\prime}$. If $\gamma=\alpha^{\prime}$ then $X=(\beta\times
K_{\alpha})\oplus Y_{\alpha^{\prime}}$ has CB-type $(\alpha,\beta)$ and
compact rank $\gamma$, as required. For the additional part, notice that if
$\alpha^{\prime}=\gamma$ is limit, this construction gives
$\gamma\in\mathrm{Ord}_{\mathrm{SD}}(X)$; if instead we want
$\gamma\in\mathrm{Ord}_{\mathrm{NSD}}(X)$, then we use $X=\beta\times
Z_{\alpha^{\prime}}$, which has still CB-type $(\alpha,\beta)$ and compact
rank $\gamma$ but is such that $\gamma\in\mathrm{Ord}_{\mathrm{NSD}}(X)$, as
desired. The remaining cases $\gamma<\alpha^{\prime}$ are treated similarly:
we set $X=(\beta\times K_{\alpha})\oplus Y_{\gamma}$ if $\gamma$ is successor
or if $\gamma$ is limit and we want $\gamma\in\mathrm{Ord}_{\mathrm{SD}}(X)$,
while we set $X=(\beta\times K_{\alpha})\oplus Z_{\gamma}$ if $\gamma$ is
limit and we want $\gamma\in\mathrm{Ord}_{\mathrm{NSD}}(X)$.

(2) Consider the space
$\hat{\mathcal{C}}={}^{\omega}\\{0,2\\}\subseteq\mathcal{N}$, which is
trivially homeomorphic to $\mathcal{C}$, and recall the spaces $Y_{\delta}$
and $Z_{\delta}$ from part (1). If $\gamma<\alpha$, then we first
build131313Here we really mean the union of the two spaces, which overlap on
the point $(0)^{\infty}$. $\hat{\mathcal{C}}_{\alpha}=\hat{\mathcal{C}}\cup
K_{\alpha+1}$, so that
$||\hat{\mathcal{C}}_{\alpha}||_{\operatorname{CB}}=\alpha$ and
$\operatorname{ker}_{\operatorname{CB}}(\hat{\mathcal{C}}_{\alpha})=\hat{\mathcal{C}}$
is compact, and then set $X=\hat{\mathcal{C}}_{\alpha}\oplus Y_{\gamma}$ or
$X=\hat{\mathcal{C}}_{\alpha}\oplus Z_{\gamma}$ depending on whether we want
$\gamma\in\mathrm{Ord}_{\mathrm{SD}}(X)$ or not in case $\gamma$ is limit. If
instead $\gamma=\alpha$, then we set $X=\hat{\mathcal{C}}\oplus Y_{\gamma}$ or
$X=\hat{\mathcal{C}}\cup Z_{\gamma}$, again depending on the given requirement
on selfduality of $\gamma$ when it is limit. (Notice that
$||\hat{\mathcal{C}}\cup Z_{\gamma}||_{\operatorname{CB}}=\gamma$ because
$(0)^{\infty}$ is no longer isolated in
$D^{\gamma}_{\operatorname{CB}}(\hat{\mathcal{C}}\cup
Z_{\gamma})=\hat{\mathcal{C}}$.)

(3) Set $X=\mathcal{N}\oplus\hat{\mathcal{C}}_{\alpha}$, where
$\hat{\mathcal{C}}_{\alpha}$ is as in part (2). ∎

##  5\. Proof of 2

The Wadge hierarchy $\mathscr{W}_{X}$ over a (zero-dimensional) Polish space
provides the finest possible measurement of the topological complexity of its
subsets. Slightly changing our perspective, it is natural to ask whether it is
possible to classify zero-dimensional Polish spaces up to the measurement
scale $\mathscr{W}_{X}$ they give rise to. More formally, consider the
standard Effros-Borel space $F(\mathcal{N})$ of all closed subsets of
$\mathcal{N}$, which up to homeomorphism contains all zero-dimensional Polish
spaces, and for $X,Y\in F(\mathcal{N})$ set

$X\approx_{\mathsf{W}}Y\iff\mathscr{W}_{X}\cong\mathscr{W}_{Y}.$

######  Remark 5.1.

The current formulation of the problem is within our axiomatic setup ${\sf
ZF}+{\sf DC}(\mathbb{R})+{\sf AD}$. If one wants to dispense with any
determinacy assumption, the problem could be formulated by requiring that
$\mathscr{W}_{X}$ and $\mathscr{W}_{Y}$ be isomorphic when restricted to Borel
sets, or even just to sets in $\boldsymbol{\Delta}^{0}_{2}$: all results we
are going to present would be unaffected by such a change, thus providing
determinacy-free anti-classification results.

The relation $\approx_{\mathsf{W}}$ is strictly coarser than closed
biembeddability: by Lemma 3.1, if $X$ and $Y$ are such that there are closed
embeddings from each space into the other one, then $X\approx_{\mathsf{W}}Y$;
but on the other hand $\mathcal{N}\approx_{\mathsf{W}}\omega\times\mathcal{C}$
even though $\mathcal{N}$ is not homeomorphic to a closed subset of
$\omega\times\mathcal{C}$.

Our analysis provides a quite satisfactory solution to the classification
problem associated with $\approx_{\mathsf{W}}$ by providing a natural
assignment of complete invariants. More precisely, call a pair of ordinals
$(\alpha,\beta)$ a _Wadge invariant_ if it satisfies the following conditions:

  * •

$\alpha,\beta\in\omega_{1}\cup\\{\Theta\\}$ and $\alpha\leq\beta$;

  * •

if $\beta\neq\Theta$, then $\beta$ is at least $2$ and is a successor ordinal;

  * •

if $\alpha\neq\Theta$, then either $\alpha=0$, or $\alpha=\lambda+1$ for some
countable limit ordinal $\lambda$, or else $\alpha=\lambda$ for some countable
ordinal $\lambda$ which is a limit of limit ordinals.

Notice that each Wadge invariant can easily be coded into a single countable
ordinal. For each zero-dimensional Polish space $X$, the pair
$(\alpha_{X},\Theta_{X})$ is a Wadge invariant which completely determines the
structure of $\mathscr{W}_{X}$ up to order-isomorphism (by the alternating
property, Proposition 4.7, and Corollary 4.14). This means that the map
$X\mapsto(\alpha_{X},\Theta_{X})$ is an assignment of complete invariants for
the classification of zero-dimensional Polish spaces up to
$\approx_{\mathsf{W}}$.

######  Remark 5.2.

As briefly discussed in Footnote 12 on page 12, in the ${\sf AD}$-world we
must make sure that when $(\alpha_{X},\Theta_{X})=(\alpha_{Y},\Theta_{Y})$
then we can actually find an isomorphism between $\mathscr{W}_{X}$ and
$\mathscr{W}_{Y}$ without appealing to the Axiom of Choice ${\sf AC}$. But
this is granted by the fact that by Theorem 2.9(2), Corollary 3.19, and Lemma
3.4, if $X\subseteq\mathcal{N}$ is closed and $\rho\colon\mathcal{N}\to X$ is
a retraction then the map $\iota_{X}$ defined by
$\boldsymbol{\Gamma}\mapsto\rho^{-1}(\boldsymbol{\Gamma})$ is an embedding of
$\langle\mathrm{Class}_{\mathsf{W}}(X),{\subseteq}\rangle$ into
$\langle\mathrm{Class}_{\mathsf{W}}(\mathcal{N}),{\subseteq}\rangle$ sending
$\mathrm{Class}_{\operatorname{NSD}}(X)$ into a suitable initial segment of
$\mathrm{Class}_{\operatorname{NSD}}(\mathcal{N})$. By our analysis, it then
turns out that $\iota_{Y}^{-1}\circ\iota_{X}$ is the required canonical
isomorphism between $\langle\mathrm{Class}_{\mathsf{W}}(X),{\subseteq}\rangle$
and $\langle\mathrm{Class}_{\mathsf{W}}(Y),{\subseteq}\rangle$.

Conversely, by Proposition 4.16 together with the computation of $\alpha_{X}$
provided after Corollary 4.14 and the computation of the length of
$\mathscr{W}_{X}$ from (the proof of) Theorem 4.15, we get a zero-dimensional
Polish space $X$ with $(\alpha_{X},\Theta_{X})=(\alpha,\beta)$ for every Wadge
invariant $(\alpha,\beta)$. In particular, the
$\approx_{\mathsf{W}}$-equivalence classes can be well-ordered in order type
$\omega_{1}$.

In contrast to these “positive” classification results, we are now going to
show that, except for the case when $X$ is a singleton, it is difficult to
determine whether $X$ belongs to a given $\approx_{\mathsf{W}}$-equivalence
class: given a Wadge invariant $(\alpha,\beta)\neq(0,2)$, the set of zero-
dimensional Polish spaces $X$ with $(\alpha_{X},\Theta_{X})=(\alpha,\beta)$ is
never Borel. This proves 2.

######  Theorem 5.3 (${\sf AD}$).

  1. (1)

The class $[\mathcal{N}]_{\approx_{\mathsf{W}}}$ corresponding to the Wadge
invariant $(\Theta,\Theta)$ is a Borel complete $\boldsymbol{\Sigma}^{1}_{1}$
set.

  2. (2)

For any $X\in F(\mathcal{N})$ with $|X|>1$ and
$X\not\approx_{\mathsf{W}}\mathcal{N}$, the set $[X]_{\approx_{\mathsf{W}}}$
is Borel $\boldsymbol{\Pi}^{1}_{1}$-hard.

  3. (3)

For every countable141414For completeness, one might ask how complex is the
set $\\{X\in F(\mathcal{N})\mid\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)\\}$ for
a given $\alpha\geq\omega_{1}$ of countable cofinality. But this is precisely
the set $[\mathcal{N}]_{\approx_{\mathsf{W}}}$ from Theorem 5.3(1). limit
ordinal $\alpha$, the set $\\{X\in
F(\mathcal{N})\mid\alpha\in\mathrm{Ord}_{\mathrm{SD}}(X)\\}$ is Borel
$\boldsymbol{\Sigma}^{1}_{1}$-hard.

As the proof will show, the same is true if we restrict the ambient space to
the analytic space $F_{>\aleph_{0}}(\mathcal{N})$ of _uncountable_ zero-
dimensional Polish spaces, i.e. to spaces with Wadge invariant of the form
$(\alpha,\Theta)$. (Just replace the Borel map $F$ with $F_{\mathcal{C}}$
everywhere.)

###### Proof.

We use the following variation of a construction from [CMMR18]. Fix a
bijection
$\langle\cdot,\cdot\rangle\colon\omega^{2}\to\omega\setminus\\{0\\}$. For
$s\in{}^{<\omega}(\omega\setminus\\{0\\})$ let $\pi(s)=t$ if and only if
$\operatorname{lh}(t)=\operatorname{lh}(s)$ and for all
$i<\operatorname{lh}(s)$ there is $n_{i}\in\omega$ such that $s(i)=\langle
t(i),n_{i}\rangle$. Define the Borel map $F\colon\mathrm{Tr}\to
F(\mathcal{N})$ from the Polish space $\mathrm{Tr}$ of trees on $\omega$ into
the space of (codes for) zero-dimensional Polish spaces by letting $F(T)$ be
the body of the pruned tree

$\\{s{{}^{\smallfrown}}(0)^{k}\mid
s\in{}^{<\omega}(\omega\setminus\\{0\\})\wedge\pi(s)\in T\wedge
k\in\omega\\}.$

If $x\in\mathcal{N}$ is an infinite branch through $T$, then $F(T)$ contains a
closed set homeomorphic to $\mathcal{N}$, namely, the set of all
$y\in\mathcal{N}$ such that $y\restriction
n\in{}^{<\omega}(\omega\setminus\\{0\\})$ and $\pi(y\restriction