# Eventually Constant and stagnating functions in non-Lindelöf spaces

Mathieu Baillif

###### Abstract

Inspired by recent work of A. Mardani which elaborates on the elementary fact
that for any continuous function $f:\omega_{1}\times\mathbb{R}\to\mathbb{R}$,
there is an $\alpha\in\omega_{1}$ such that
$f(\langle\beta,x\rangle)=f(\langle\alpha,x\rangle)$ for all $\beta\geq\alpha$
and $x\in\mathbb{R}$, we introduce four properties $\mathsf{P}(X,Y)$,
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{S},\mathsf{L},\mathsf{BR}\\}$, which are
different formalizations of the idea vaguely stated as “given a continuous
$f:X\to Y$, there is a small subspace of $X$ outside of which $f$ does not do
anything much new”. More precisely, we say that the spaces $X,Y$ satisfy the
property $\mathsf{EC}(X,Y)$ (resp. $\mathsf{S}(X,Y)$) [resp.
$\mathsf{L}(X,Y)$] iff given $f:X\to Y$, then there is a Lindelöf $Z\subset X$
such that $f(X-Z)$ is a singleton (resp. there is a retraction $r:X\to Z$ such
that $f\circ r=f$) [resp. $f(Z)=f(X)$]. ($\mathsf{BR}(X,Y)$ is defined
similarly.) We investigate the relations between these four and other
classical topological properties. Actually, two variants
$\mathsf{P},\mathsf{P_{cl}}$ of each property are given depending on whether
$Z$ can be chosen to be closed. To get an idea of what our results look like,
here is a sample.
An uncountable subspace $T$ of a tree of height $\omega_{1}$ is
$\omega_{1}$-compact iff $\mathsf{S}(T,Y)$ holds for any metrizable space $Y$
of cardinality $>1$. (The case $Y=\mathbb{R}$ and $T$ a Suslin tree was proved
by Steprāns long ago.) If $M$ is a $\aleph_{1}$-strongly collectionwise
Hausforff non-metrizable manifold satisfying either (a weakening of)
$\mathsf{S}(M,\mathbb{R})$ or $\mathsf{EC}(M,\mathbb{R})$, then $M$ is
$\omega_{1}$-compact. The property $\mathsf{L}(M,\mathbb{R})$ holds for any
manifold while $\mathsf{L}(M,\mathbb{R}^{2})$ does not. Under PFA, a locally
compact countably tight space $Y$ for which $\mathsf{EC}(\omega_{1},Y)$ holds
is isocompact, while there are counterexamples under $\clubsuit_{C}$. Some of
our results are (more or less elaborate) restatements of other researchers
work put in our context.

###### Contents

  1. 1 Introduction
  2. 2 Does a lazy constant broken record stagnate ?
  3. 3 Consecuences of $\mathsf{EC}(X,\mathbb{R})$; $\mathsf{EC}(\omega_{1},Y)$ and isocompactness of $Y$
  4. 4 $\mathsf{S}(T,\mathbb{R})$ and Suslin trees
  5. 5 $\mathsf{BR}(X,Y)$, $\mathsf{L}(X,Y)$ when $Y$ is metric and $X$ an increasing chain of ‘nice’ subspaces
  6. 6 A word on products on the target space
  7. 7 Type I manifolds: when $\mathsf{EC}\not\Rightarrow\mathsf{S}$.
  8. 8 $w\mathsf{S}(M,\mathbb{R})$, normality, collectionwise Hausdorfness and $\omega_{1}$-compactness for manifolds
  9. 9 If ‘small’ means ‘compact’
  10. 10 Summary and tables

##  1 Introduction

In this note, by ‘space’ we mean topological Hausdorff space (in particular
‘regular’ and ’normal’ imply Hausdorff) and every function is assumed
continuous, unless specified. We denote ordered pairs with brackets
$\langle\cdot,\cdot\rangle$, reserving parenthesis for intervals in (totally)
ordered spaces. We refer to [10] for standard topological notions not defined
in our text. Notice however that we sometimes depart from the conventions of
[10] about including or not separation axioms into properties like
pseudocompactness. We will specify when this is the case.
Our starting point is the following well known elementary result. (For a
proof, combine Lemma B.30 in [11] with the fact that $\mathbb{R}$ is
separable.) As usual $\omega_{1}$ is the first uncountable ordinal endowed
with the order topology.

######  Theorem 1.1.

Let $f:\omega_{1}\times\mathbb{R}\to\mathbb{R}$ be continuous. Then there is
$\alpha\in\omega_{1}$ such that
$f(\langle\beta,x\rangle)=f(\langle\alpha,x\rangle)$ for all $\beta\geq\alpha$
and $x\in\mathbb{R}$.

To summarize in a very imprecise catchphrase: Outside of a small subspace, $f$
does not do anything much new.111Another phrasing is $f$ does not cause much
ruckus outside of a very small world. We feel that this phrase can be also
seen an an overstatement of this article’s fate. Here of course the small
subspace is $[0,\alpha]\times\mathbb{R}$. The purpose of this note is to
investigate the notions obtained when ‘small’ and ‘nothing much new’ are
interpreted in various ways. Well, not so various as to the former: ‘Small’
will almost always mean ‘Lindelöf’ (the only exception being the last
section). This might be surprising at first since compactness is arguably a
more natural mesure of smallness of a subspace. But we believe that our choice
yields more interesting results, as replacing it by compactness as we do in
Section 9 seems to (almost always) confine us to pseudocompact spaces, at
least for real valued maps.
The phrase ‘$f$ does not do anything much new’ gives way to more varied
interpretations (at least to the extent permitted by our imagination). A
rather general interpretation is that a there is a small subset whose image is
equal to that of the whole domain space. In Theorem 1.1,
$f([0,\alpha]\times\mathbb{R})=f(\omega_{1}\times\mathbb{R})$. Figuratively
speaking, $f$ (or $\omega_{1}\times\mathbb{R}$) is a lazy explorer of
$\mathbb{R}$: after a small exploration it remains at the same places
forever.222We could have invoked the figure of a regular customer in a western
european local bar, since these individuals tend to keep to the same place as
well after choosing their favorite table and chair. We refrained from it in
view of the confusion induced by the terminology collision with regular
spaces, and by broken-record-like conversations which tend to emerge in these
settings, which would mistakenly allude to the property defined just next.
Looking at it from the other side, we can say that outside of a small subset,
any value taken by $f$ will be taken again and again, like a broken record, as
we go further away (horizontally in this case) in the domain space. In Theorem
1.1, if $\beta>\alpha$, there is $\gamma>\beta$ such that
$f(\langle\gamma,t\rangle)=f(\langle\beta,t\rangle)$ (this actually holds for
each $\gamma$). Another more restrictive (a priori) interpretation is that the
map $f$ stagnates outside of a small subspace; that is, there is a retraction
$r$ of $X$ onto a small subset such that $f\circ r=f$. A retraction is a map
$r:X\to Z\subset X$ whose restriction to $Z$ is the identity. In Theorem 1.1
we may define it as
$r(\langle\beta,t\rangle)=\langle\min\\{\alpha,\beta\\},t\rangle$. Lastly,
looking only at horizontal lines, we see that any $g:\omega_{1}\to\mathbb{R}$
is eventually constant, in the sense that $g$ is constant outside of the small
subset $[0,\alpha]$.
These four interpretations are formalized in the following definition.

######  Definition 1.2.

Let $X,Y$ be spaces with $X$ non-Lindelöf. Then $X,Y$ satisfy the left side of
the table below iff for each $f:X\to Y$, there is a Lindelöf subset $Z\subset
X$ such that the right side holds. The middle boxes contain the shorthands for
each property.

$X$ is eventually constant in $Y$ | $\mathsf{EC}(X,Y)$ |  $f(X-Z)$ is a singleton
---|---|---
$X$ stagnates in $Y$ | $\mathsf{S}(X,Y)$ | there is a retraction
$r:X\to Z$ satisfying $f=f\circ r$
$X$ is a lazy explorer of $Y$ | $\mathsf{L}(X,Y)$ | $f(Z)=f(X)$
$X$ is a broken record in $Y$ | $\mathsf{BR}(X,Y)$ |  for each Lindelöf $W\supset Z$,
$f(X-W)=f(X-Z)$

If $Z$ can be chosen to be closed, we write $\mathsf{EC_{cl}}(X,Y)$, etc, for
the stronger properties.

(We already stress that there are spaces $X$ for which, really, nothing much
new happens outside of some closed Lindelöf subset for any real valued map,
but none of these properties with $Y=\mathbb{R}$ hold, see Example 2.10 (c).)
Notice in passing that any space $X$ with the property that any real valued
fonction on $X$ is constant satisfies $\mathsf{P_{cl}}(X,\mathbb{R})$ for each
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{S},\mathsf{L},\mathsf{BR}\\}$. Such
spaces do exist, some being even regular (for instance [29, Example 92]).
Spaces $Y$ satisfying $\mathsf{EC}(\omega_{1},Y)$ are called
$\omega_{1}$-squat by A. Mardani, whose PhD thesis [17] contains results about
this and related classes of spaces which motivated this work. The term ‘squat’
was first used by D. Gauld (in particular in [11]) for almost the same
property. This note contains our musings about the interplays these eight
notions have with each other and other topological properties in various
classes of spaces. It is organized in sections of variable length whose titles
hint to their contents. We were driven by pure curiosity and the pleasure of
wandering in this landscape333We believe that it is important for the reader
to keep that in mind. Despite the length of this paper, we have a leisurely
approach and do not claim any grand result.. In some cases we have tried to
obtain quite general results (for instance in sections 2–3), while in others
we have concentrated our efforts on particular classes of spaces, such as set-
theoretic trees (the entire section 4) and non-metrizable manifolds (all of
sections 7–8 and some of section 5). As such, this note is more akin to a
little stroll in the garden, with rusty old tools in hands, peeking below
scattered rocks and looking for strange insects; than to securing the
foundations for a (future) twenty storeys tower. We do not know if our results
will seem appealing to other researchers who tend to enjoy the scenery
differently from us, but we hope that the reader’s mood is bucolic enough to
at least enjoy some of them. The sections are somewhat independent, although
the later ones tend to refer to the previous ones (what a surprise). Section 2
contains basic results used pervasively, and section 10 contains tables which
briefly summarize the properties of the strange insects we encountered.

We end this introduction by recalling definitions that are either not-so-
standard or slightly different from the usual ones.

By cover of a space $X$ is understood a family of open sets whose union
contains $X$. A cover is a chain cover if it is linearly ordered by the
inclusion relation. Any non-trivial chain cover (that is, one without the
whole space as a maximal element) has a subcover indexed by a regular cardinal
whose members are pairwise distinct. We always implicitely take such a
subcover and use such indexing. A space $X$ is of Type I (Nyikos [20]) iff
$X=\cup_{\alpha<\omega_{1}}X_{\alpha}$, where $X_{\alpha}$ is open,
$\overline{X_{\alpha}}\subset X_{\beta}$ whenever $\alpha<\beta$, and
$\overline{X_{\alpha}}$ is Lindelöf for each $\alpha$. Although it it not
included in the usual definition, for simplicity we assume in this note that
Type I spaces are not Lindelöf, that is, $X\not=X_{\alpha}$ for each $\alpha$.
If $\cup_{\alpha<\beta}X_{\alpha}=X_{\beta}$ whenever $\beta$ is limit, the
cover $\\{X_{\alpha}\,:\,\alpha<\omega_{1}\\}$ is called canonical. Any chain
cover of a Type I space has a subcover indexed by $\omega_{1}$ which can be
made canonical by adding the missing $X_{\alpha}$’s. Any two canonical covers
agree on a club (i.e. closed and unbounded) subset of $\omega_{1}$, as easily
seen. Hence, if $X$ is a Type I space, $X_{\alpha}$ will always denote the
$\alpha$th member of some (often implicit) canonical cover. We borrow the
usual vocabulary used in $\omega_{1}$ for Type I spaces: a subset of $X$ is
bounded iff it is contained in some $X_{\alpha}$ and unbounded otherwise, and
club means closed and unbounded. The closure of a Lindelöf subset is Lindelöf
in a Type I space.

By manifold or $n$-manifold (when we want to emphasize the dimension) we mean
a connected space locally homeomorphic to $\mathbb{R}^{n}$. A surface is a
$2$-manifold. Connectedness is often an indispensable property in our results
about manifolds. Manifolds with boundary have some points (those in the
manifold boundary) with open neighborhoods homeomorphic to $\mathbb{R}_{\geq
0}\times\mathbb{R}^{n-1}$ (and no open neighborhood homeomorphic to
$\mathbb{R}^{n}$). Unless specified, the word ‘boundary’ alone means
‘topological boundary’ and not ‘manifold boundary’. The longray
$\mathbb{L}_{\geq 0}$ is the $1$-manifold with boundary
$\omega_{1}\times[0,1)$ with lexicographic order topology (not the product
topology). We often view $\omega_{1}$ as a subset of $\mathbb{L}_{\geq 0}$ by
identifying $\alpha$ with $\langle\alpha,0\rangle$, and hence write for
instance $\mathbb{L}_{\geq_{0}}-\omega_{1}$ instead of
$\mathbb{L}_{\geq_{0}}-\omega_{1}\times\\{0\\}$. The open longray
$\mathbb{L}_{+}$ is obtained by deleting the $0$ point in $\mathbb{L}_{\geq
0}$. A space is $\omega$-bounded iff any countable subset has a compact
closure. For Type I spaces this is equivalent to being countably compact. A
manifold is $\omega$-bounded iff it is countably compact and of Type I [11,
Thm 4.10]. A longpipe [11, Def. 4.11] is an $\omega$-bounded (hence Type I)
surface $S=\cup_{\alpha\in\omega_{1}}S_{\alpha}$ such that $S_{\alpha+1}$ is
homeomorphic to the cylinder $\mathbb{S}^{1}\times[0,1)$ and the topological
boundary of $S_{\alpha+1}$ in $S_{\beta}$ is homeomorphic to the circle for
each $\alpha$ and $\beta\geq\alpha+2$. (This may not be true if $\alpha$ is a
limit ordinal, though.) For more on non-metrizable manifolds and longpipes,
see [11] and [20].

Finally, recall that a space is $\omega_{1}$-compact or has countable spread
iff its closed discrete subspaces are at most countable. The following
classical facts (see e.g. [10, Thm 4.1.15, Thm 4.1.17]) will be useful in some
proofs.

######  Lemma 1.3.

In a metrizable space, the properties $\omega_{1}$-compact, Lindelöf,
hereditarily Lindelöf, separable and hereditarily separable are all
equivalent, and compactness is equivalent to countable compactness.

##  2 Does a lazy constant broken record stagnate ?

An astute reader probably suspects that the answer to the question in this
section’s title is absolutely yes, but sometimes no, and reciprocally. This
section is devoted to giving more details about this answer (and the question
itself). When there is no risk of confusion, we abbreviate the statement ‘for
all Hausdorff spaces $X,Y$,
$\mathsf{P}_{1}(X,Y)\Rightarrow\mathsf{P}_{2}(X,Y)$’ by
‘$\mathsf{P}_{1}\Rightarrow\mathsf{P}_{2}$’.
‘$\mathsf{P}_{1}\not\Rightarrow\mathsf{P}_{2}$’ and
‘$\mathsf{P}_{1}\Leftrightarrow\mathsf{P}_{2}$’ are to be understood
similarly. Notice that for $\mathsf{S}(X,Y)$, $Z$ has to be closed in
Definition 1.2 since it is the set of fixed points of the retraction and the
spaces are Hausdorff, so $\mathsf{S}\Leftrightarrow\mathsf{S_{cl}}$. The
others implications in the lemma below are immediate from the definitions.
(For aesthetical reasons, we tend to denote implication by single arrows
$\longrightarrow$ in somewhat complicated diagrams, and by double arrows
$\Longrightarrow$ in one-liners formulas. We hope that it does not cause
confusion.)

######  Lemma 2.1.

The following implications hold.

${\mathsf{EC_{cl}}}$${\mathsf{BR_{cl}}}$${\mathsf{BR}}$${\mathsf{EC}}$${\mathsf{L_{cl}}}$${\mathsf{S_{cl}}}$${\mathsf{S}}$${\mathsf{L}}$

Implications not shown actually do not hold for all spaces $X,Y$. In
particular, none of $\mathsf{L}$, $\mathsf{EC}$, $\mathsf{BR}$ imply their
$\mathsf{cl}$-counterpart in general. It is however the case for Type I
spaces.

######  Lemma 2.2.

For Type I spaces $X$, $\mathsf{P}(X,Y)\Leftrightarrow\mathsf{P_{cl}}(X,Y)$
for any space $Y$ and
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{L},\mathsf{BR},\mathsf{S}\\}$.

###### Proof.

For $\mathsf{S}$, this always holds regardless of whether $X$ is Type I.
Notice that in the definitions of properties
$\mathsf{EC},\mathsf{L},\mathsf{BR}$ in 1.2, if the right side holds for the
Lindelöf subset $Z$ then it holds for any Lindelöf subset containing $Z$. If
$X$ is Type I, a Lindelöf subspace of $X$ is contained in
$\overline{X_{\alpha}}$ (Lindelöf itself) for some $\alpha$, which yields the
result. ∎

Let us now have a look at arrows not in Lemma 2.1. Our first example is
trivial.

######  Example 2.3.

None of $\mathsf{S_{cl}},\mathsf{L_{cl}},\mathsf{BR_{cl}}$ imply
$\mathsf{EC}$: A space $X$ consisting of the disjoint union of two copies of
$\omega_{1}$ satisfies the first three properties for real maps but not the
last one.

###### Details.

By Theorem 1.1. Taking a map sending one copy of $\omega_{1}$ on $0$ and the
other on $1$ shows that $\mathsf{EC}(X,\mathbb{R})$ does not hold. ∎

Other examples as 2.3 are the long line $\mathbb{L}$ made of two copies of
$\mathbb{L}_{\geq 0}$ glued at their $0$-point, and the space
$\omega_{1}\times\mathbb{R}$.

We now show that $\mathsf{P}\not\Rightarrow\mathsf{P_{cl}}$ for
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{L},\mathsf{BR}\\}$. The idea is actually
quite simple and can be infered from the two next lemmas. The first one is
trivial.

######  Lemma 2.4.

Let $\mathsf{P}\in\\{\mathsf{EC},\mathsf{L},\mathsf{BR}\\}$ and $X$, $Y$ be
spaces such that $X=A\cup B$ where $A$ is Lindelöf and $\mathsf{P}(B,Y)$
holds. Then $\mathsf{P}(X,Y)$ holds.

######  Lemma 2.5.

Let $X$ be a space with a dense countable subspace of isolated points which we
identify with the integers $\omega$. If $X-\omega$ is not Lindelöf, then
$\mathsf{L_{cl}}(X,\mathbb{R})$ and $\mathsf{BR_{cl}}(X,\mathbb{R})$ do not
hold.

###### Proof.

Let $f:X\to\mathbb{R}$ be defined by $f(n)=1/n$ for $n\in\omega$ and $f(x)=0$
for $x\not\in\omega$. Then $f$ is continuous. Any $Z\subset X$ such that
$f(Z)=f(X)$ must contain all of $\omega$, hence its closure is all of $X$,
which is not Lindelöf. It follows that $\mathsf{L_{cl}}(X,\mathbb{R})$ does
not hold. The same function shows that $\mathsf{BR_{cl}}(X,\mathbb{R})$ does
not hold: given $Z\subset X$, if $\omega\not\subset Z$, for any $x\in\omega-Z$
we have

$f(x)\in f(X-Z)\not=f(X-(Z\cup\\{x\\}))\not\ni f(x).$

Hence, $Z\supset\omega$ and its closure must again be the entire space. ∎

Hence, spaces as in this lemma are good candidates to show that
$\mathsf{P}\not\rightarrow\mathsf{P_{cl}}$. To complete the task, the
following result is another tool. Recall that $\beta X$ is the Čech-Stone
compactification of the space $X$.

######  Theorem 2.6.

Let $X$ be Tychonoff and let $\beta X$ be its Čech-Stone compactification.
(a) If $|\beta X-X|=1$, then $\mathsf{EC}(X,\mathbb{R})$ holds.
(b) If $\beta X-X$ is at most countable and $X$ is locally compact, then
$\mathsf{L}(X,\mathbb{R})$ holds.

Recall that a $0$-set (resp. a co-$0$-set) in a space $X$ is a preimage of
$\\{0\\}$ (resp. of $(0,1]$) under a map $X\to[0,1]$.

###### Proof.

If $A,B$ are disjoint $0$-sets in $X$, then their closure is disjoint in
$\beta X$. (All claimed properties of $\beta X$ in this proof can be found in
[10, Section 3.6].) This implies that $|\beta X-X|=1$ is equivalent to the
property that given two disjoint $0$-sets in $X$, then at least one is
compact. We show in Theorem 3.3 below that this implies
$\mathsf{EC}(X,\mathbb{R})$, which proves (a).
Assume as in (b) that $\beta X-X$ is at most countable, with $X$ locally
compact, and let $f:X\to\mathbb{R}$ be given. We may assume that the range of
$f$ is contained in $[0,1]$. Let $\beta f:\beta X\to[0,1]$ be the extension of
$f$ to all of $\beta X$ (which always exists, see e.g. [10, Theorem 3.6.1]).
By local compactness, $X$ is open in $\beta X$ (see e.g. [10, Theorem 3.5.8]).
Then $C=\beta f(\beta X-X)$ is compact and countable in $[0,1]$. If
$E\subset[0,1]$ is closed and disjoint from $C$, then $f^{-1}(E)$ is compact
in $X$, otherwise its closure in $\beta X$ intersects $\beta X-X$, and hence
$E\cap C\not=\varnothing$. Since any open subset of $[0,1]$ is an
$F_{\sigma}$, $Z_{0}=f^{-1}([0,1]-C)$ is Lindelöf. Add to $Z_{0}$ one preimage
of each $c\in C$ such that $f^{-1}(C)\cap X\not=\varnothing$, to obtain a
Lindelöf subset $Z$ with full image. ∎

######  Example 2.7 (S. Mrówka, in effect).

There is a locally compact first countable pseudocompact separable space $X$
such that:
(a) $\mathsf{EC}(X,\mathbb{R})$ holds and hence so do
$\mathsf{L}(X,\mathbb{R})$ and $\mathsf{BR}(X,\mathbb{R})$,
(b) $\mathsf{L_{cl}}(X,\mathbb{R})$ does not hold and thus neither do
$\mathsf{EC_{cl}}(X,\mathbb{R})$ and $\mathsf{S}(X,\mathbb{R})$,
(c) $\mathsf{BR_{cl}}(X,\mathbb{R})$ does not hold.

A space $X$ is pseudocompact iff every real valued map defined on $X$ has a
bounded image. We do not assume that pseudocompact spaces are Tychonoff,
departing from the definition given in [10, p. 208]. See e.g. [30] for more on
pseudocompact spaces. Recall also that a family of sets is almost disjoint iff
the intersection of any two members is finite.

###### Details.

Recall that a $\psi$-space is the union of an open countable discrete space
(which we may take to be $\omega$) and an uncountable discrete subspace whose
points are given by a maximal family $\mathcal{R}$ of almost disjoint subsets
of $\omega$. A neighborhood basis of $A\in\mathcal{R}$ is given by
$\\{A\\}\cup(A-F)$ where $F$ is finite. All $\psi$-spaces are pseudocompact
and locally compact. S. Mrówka [18, Theorem 3.11] shows how to construct a
$\psi$-space $X$ whose Čech-Stone compactification is the one point
compactification. This implies that $\mathsf{EC}(X,\mathbb{R})$ holds by
Theorem 2.6. Of course, $\mathsf{L_{cl}}(X,\mathbb{R})$ and
$\mathsf{BR_{cl}}(X,\mathbb{R})$ do not hold by Lemma 2.5. ∎

Notice in passing that a disjoint union of two spaces as in Example 2.7 yields
a space satisfying $\mathsf{L}(X,\mathbb{R})$ and $\mathsf{BR}(X,\mathbb{R})$
but neither their cl-versions nor $\mathsf{EC}(X,\mathbb{R})$. Another way of
obtaining $\mathsf{P}\not\Rightarrow\mathsf{P_{cl}}$ is the following. For the
definition of the uncountable cardinal $\mathfrak{p}$ and more on the subject,
see see e.g. [8, Chapter 3].

######  Example 2.8 (P. Nyikos).

There is a locally compact first countable separable normal space $X$ such
that $\mathsf{EC}(X,\mathbb{R})$ holds but $\mathsf{L_{cl}}(X,\mathbb{R})$ and
$\mathsf{BR_{cl}}(X,\mathbb{R})$ do not. If $\mathfrak{p}=\omega_{1}$, the
space can be made to be moreover countably compact.

###### Details.

If $X$ is as in Lemma 2.5 but such that $X-\omega$ is homeomorphic to
$\omega_{1}$ (in the order topology), then $\mathsf{EC}(X,\mathbb{R})$ holds
(by Lemma 2.4) while $\mathsf{L_{cl}}(X,\mathbb{R})$ and
$\mathsf{BR_{cl}}(X,\mathbb{R})$ do not. There are many ways to obtain normal
first countable and locally compact such spaces, see for instance [23]. Nyikos
([21], Theorem 2.1 and Example 3.4) shows how to obtain a countably compact
example whenever $\mathfrak{p}=\omega_{1}$. ∎

A variation of this space can be adapted to obtain a surface, see section 9.
The next examples show in particular that
$\mathsf{L}\not\Rightarrow\mathsf{BR}$ and
$\mathsf{BR}\not\Rightarrow\mathsf{L}$. It is convenient to first introduce
the following notation. If $f:X\to Y$ is a map with $X$ of Type I, set:

$\displaystyle\mathsf{Bd}(f)$ $\displaystyle=\\{y\in
Y\,:\,f^{-1}(\\{y\\})\text{ is Lindel\"{o}f in }X\\}.$

######  Lemma 2.9.

Let $X,Y$ be spaces. Then

$\mathsf{BR}(X,Y)\text{ holds }\Longleftrightarrow
f^{-1}(\mathsf{Bd}(f))\text{ is contained in a Lindel\"{o}f subset for any
}f:X\to Y.$

###### Proof.

Essentially by unraveling the definitions. Here are some details for the
skeptics. Suppose that $\mathsf{BR}(X,Y)$ holds. Let $f:X\to Y$ be given and
let $Z$ be Lindelöf such that for any Lindelöf $W\supset Z$, $f(X-W)=f(X-Z)$.
Let $y\in\mathsf{Bd}(f)$, then $W=f^{-1}(\\{y\\})\cup Z$ is Lindelöf. If $y$
has a preimage outside of $Z$, then $f(X-W)$ does not contain $y$ while
$f(X-Z)$ does, a contradiction. Hence $f^{-1}(\mathsf{Bd}(f))\subset Z$.
Conversely, let $f:X\to Y$ be given such that $f^{-1}(\mathsf{Bd}(f))$ is
contained in a Lindelöf subset of $X$. Hence
$f^{-1}(\mathsf{Bd}(f))\subset\overline{X_{\alpha}}$ for some $\alpha$. Let
$W\supset\overline{X_{\alpha}}$ be Lindelöf. If $y\in
f(X-\overline{X_{\alpha}})$, then $f^{-1}(\\{y\\})$ is unbounded, and hence
$f^{-1}(\\{y\\})\not\subset W$. It follows that $y\in f(X-W)$. This shows that
$\mathsf{BR}(X,Y)$ holds. ∎

######  Example 2.10.

There are Type I locally metrizable spaces $H^{-}$ and $H^{+}$ such that:
(a) $\mathsf{S_{cl}}(H^{-},\mathbb{R})$ and thus
$\mathsf{L_{cl}}(H^{-},\mathbb{R})$ hold but $\mathsf{BR}(H^{-},\mathbb{R})$
does not,
(b) $\mathsf{BR_{cl}}(H^{+},\mathbb{R})$ holds but
$\mathsf{L}(H^{+},\mathbb{R})$ does not,
(c) $H=H^{-}\sqcup H^{+}$, the topological disjoint sum of $H^{-}$ and
$H^{+}$, has a partition into closed non-Lindelöf subsets such that for each
map $f:H\to\mathbb{R}$ there is a Lindelöf subset $Z$ such that $f$ is
constant on each member of the partition outside of $Z$ but none of
$\mathsf{P}(H,\mathbb{R})$ holds for
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{L},\mathsf{BR},\mathsf{S}\\}$.

(Recall that for Type I spaces the properties are equivalent to their
$\mathsf{cl}$-counterparts, we worded the theorem in the strongest form.)

###### Details.

The examples are simple subspaces of $\omega_{1}\times[0,1]$. Take an
uncountable subset $S=\\{s_{\alpha}\,:\,\alpha\in\omega_{1}\\}$ of $[0,1]$
with dense complement (all the $s_{\alpha}$ are distinct). Then set:

$\displaystyle H^{+}$
$\displaystyle=\bigcup_{\alpha\in\omega_{1}}[\alpha,\omega_{1})\times\\{s_{\alpha}\\}$
$\displaystyle H^{-}$ $\displaystyle=\omega_{1}\times[0,1]-H^{+}.$

(The intervals $[\alpha,\omega_{1})$ are taken in $\omega_{1}$.) Then both are
a Type I spaces with canonical covers $H^{-}_{\alpha},H^{+}_{\alpha}$ given by
the intersection of the space with $[0,\alpha)\times[0,1]$.
(a) Let $f:H^{-}\to[0,1]$ be the projection on the second coordinate. Then
$\mathsf{Bd}(f)=S$ and $f^{-1}(\mathsf{Bd}(f))$ is unbounded (it contains
$[0,\alpha)\times\\{s_{\alpha}\\}$ for each $\alpha$); hence,
$\mathsf{BR}(H^{-},\mathbb{R})$ does not hold. To see that
$\mathsf{S}(H^{-},\mathbb{R})$ does hold, take a countable dense subset
$Q=\\{q_{n}\,:\,n\in\omega\\}$ of $[0,1]-S$. Since $S$ has dense complement,
$\overline{Q}=[0,1]$. Given a map $g:H^{-}\to\mathbb{R}$, there is $\beta$
such that $g$ is constant above $\beta$ on each horizontal
$\omega_{1}\times\\{q_{n}\\}$, and thus by density on every horizontal line,
even those who do not go the entire length. Hence the retraction
$r(\langle\alpha,t\rangle)=\langle\min\\{\alpha,\beta\\},t\rangle$ satisfies
$g\circ r=g$.
(b) The fact that $S$ has dense complement is actually irrelevant for this
part. The projection on the second coordinate shows that
$\mathsf{L}(H^{+},\mathbb{R})$ does not hold: new values are introduced as far
as one wants. To see that $\mathsf{BR}(H^{+},\mathbb{R})$ does hold, use the
same argument as in (a) with a countable dense subset
$\\{s_{\alpha_{n}}\,:n\in\omega\\}$ of $S$ to show that there is $\beta$ such
that $f$ is eventually constant above $\beta$ on every horizontal line. Take
$Z=\overline{H^{+}_{\beta}}$, then any Lindelöf $W\supset Z$ satisfies
$f(H^{+}-Z)=f(H^{+}-W)$.
(c) Notice that while $H^{+}\cup H^{-}=\omega_{1}\times[0,1]$, the topological
disjoint sum $H=H^{+}\sqcup H^{-}$ has a finer topology, as both $H^{+}$ and
$H^{-}$ are clopen in $H$. Partition $H$ into its horizontal lines. Then the
claimed properties are immediate from the arguments above. ∎

It is not possible to find a manifold having the same properties as $H^{+}$:
Theorem 5.1 implies that $\mathsf{L}(X,\mathbb{R})$ holds for any manifold
$M$. We however do not know the answer to the following question.

######  Question 2.11.

Is there a manifold $M$ such that $\mathsf{S}(M,\mathbb{R})$ holds but not
$\mathsf{BR}(M,\mathbb{R})$ ?

See section 5 for more on $\mathsf{BR}$ and sections 7–8 for more on
$\mathsf{S}$. While $\mathsf{EC}(X,Y)$ (and a fortiori
$\mathsf{EC_{cl}}(X,Y)$) seems to be a stronger property than
$\mathsf{S}(X,Y)$, it does not imply it since $X$ might lack retractions on
sufficiently large subspaces (see Example 2.8). Actually, $\mathsf{EC}(X,Y)$
does imply the negation of a property weaker than $\mathsf{S}(X,Y)$ if $X$ is
a longpipe, see Theorem 7.6 and Example 7.7. This was the only implication not
in Lemma 2.1 and not ruled out by our examples so far.

Let us end this section by getting away with general (almost) trivialities.
Firstly, the following obviously holds (as already noted in [17, Lemma 4.3.34]
for a space eventually constant in another):

######  Lemma 2.12.

Let $X,Y,Z$ be spaces such that there is a continuous $1$-to-$1$ map $Y\to Z$.
Then $\mathsf{P}(X,Z)\Longrightarrow\mathsf{P}(X,Y)$ for
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{L},\mathsf{BR},\mathsf{S}\\}$ or the
$\mathsf{cl}$ versions.

######  Corollary 2.13.

Let $\tau\supset\rho$ be Hausdorff topologies on $Y$. Then
$\mathsf{P}(X,\langle Y,\rho\rangle)\Longrightarrow\mathsf{P}(X,\langle
Y,\tau\rangle)$ for
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{L},\mathsf{BR},\mathsf{S}\\}$ or the
$\mathsf{cl}$ versions.

###### Proof.

$id:\langle Y,\tau\rangle\to\langle Y,\rho\rangle$ is a $1$-to-$1$ continuous
map. ∎

Finally, the next lemmas are also almost immediate.

######  Lemma 2.14.

Let $X=\cup_{\alpha\in\omega_{1}}X_{\alpha}$ be a Type I space and $Y$ be a
countable space with no separation axiom assumed. Then $\mathsf{L}(X,Y)$ and
$\mathsf{BR}(X,Y)$ hold.

###### Proof.

Let $f:X\to Y$ be given. Then $f(X_{\alpha})$ and $f(X-X_{\alpha})$ are
respectively increasing and decreasing $\omega_{1}$-sequences of countable
sets, they must then stagnate above some $\alpha$. ∎

######  Lemma 2.15.

Let $X$ be a space containing a clopen uncountable discrete subspace $A$ and
let $Y$ be a space. Then the following hold.
(a) If $|Y|\geq\aleph_{1}$, then neither $\mathsf{L}(X,Y)$ nor
$\mathsf{BR}(X,Y)$ do hold.
(b) If $|Y|\geq 2$, then neither $\mathsf{EC}(X,Y)$ nor $\mathsf{S}(X,Y)$ do
hold.

###### Proof.

(a) By assumption, we can define a map $f:X\to Y$ which is $1$-to-$1$ on $A$
and constant on $X-A$. Then $f$ contradicts $\mathsf{L}(X,Y)$ and
$\mathsf{BR}(X,Y)$.
(b) Partition $A$ into two clopen discrete uncountable subsets $A_{0},A_{1}$.
Define $f:X\to Y$ that sends $A_{0}$ to one of the points of $Y$ and $S-A_{0}$
to the other one. This defines a continuous map which contradicts
$\mathsf{S}(S,Y)$ and $\mathsf{EC}(S,Y)$ ∎

##  3 Consecuences of $\mathsf{EC}(X,\mathbb{R})$; $\mathsf{EC}(\omega_{1},Y)$
and isocompactness of $Y$

Theorem 1.1 shows in particular that $\mathsf{EC}(\omega_{1},\mathbb{R})$
holds. In this section, we first investigate which spaces satisfy
$\mathsf{EC}(X,\mathbb{R})$, and when does $\mathsf{EC}(X,\mathbb{R})$ imply
$\mathsf{EC}(X,Y)$. (Our results have similarities with those in [24, Section
7].) Then, we look for properties of $Y$ that imply or are implied by
$\mathsf{EC}(Z,Y)$ when $Z$ is similar (in some way) to $\omega_{1}$. The
notions of C-closedness and isocompactness will be central (see definitions
below).

It is well known and easy to prove that finite unions and at most countable
intersections of $0$-sets are $0$-sets, and finite intersections and countable
unions of co-$0$-sets are co-$0$-sets. If $Y$ is perfectly normal (in
particular, metric), then the preimage of a closed subset of $Y$ is a $0$-set.

######  Lemma 3.1.

Let $X$ be a space, and $E\subset D\subset X$ be subspaces such that $E$ is
Lindelöf and $D$ is non-Lindelöf. Then the following hold.
(a) There is an open $U\supset E$ such that $D-U$ is non-Lindelöf.
(b) If $X$ is Tychonoff and $D$ is a $0$-set, then there is an open $U\supset
E$ such that $D-U$ is a non-Lindelöf $0$-set.

###### Proof.

Since $D$ is non-Lindelöf and $E$ Lindelöf, let $\mathcal{U}$ be a cover of
$D$ without countable subcover and let $\mathcal{U}_{0}\subset\mathcal{U}$ be
a countable subcover of $E$. Then $D-\cup\mathcal{U}_{0}$ is non-Lindelöf,
which proves (a). For (b), since $X$ is Tychonoff for each $x\in E$ we may fix
$g_{x}:X\to[0,1]$ such that $g_{x}(x)=1$ and $g_{x}$ is $0$ outside of
$\cup\mathcal{U}_{0}$. Let $\\{x_{n}\,:\,n\in\omega\\}$ be such that
$\mathcal{W}=\\{g_{x_{n}}^{-1}((0,1])\,:\,n\in\omega\\}$ is a cover of $E$.
Then $\cup\mathcal{W}$ is a countable union of co-$0$-sets and hence a
co-$0$-set and is included in $\cup\mathcal{U}_{0}$. It follows that
$D-\cup\mathcal{W}$ is a $0$-set which is non-Lindelöf since it contains the
closed non-Lindelöf subset $D-\cup\mathcal{U}_{0}$. ∎

######  Definition 3.2.

We say that a space $X$ has property $\mathsf{IC}$ (resp. $\mathsf{I0}$) iff
given two disjoint closed subsets (resp. $0$-sets) of $X$, at least one of
them is Lindelöf.

The following theorem was essentially proved for Type I spaces in the preprint
[3, Thm 6.1 and Lemma 6.2].

######  Theorem 3.3.

Let $X$ be a space. Then the properties below are related as follows.

(1)$\Longleftrightarrow$(2)$\Longleftrightarrow$(3a)$\Longleftarrow$(3b)$\Longleftarrow$(4a)$\Longleftrightarrow$(4b)

Moreover, if $X$ is Tychonoff then (3a)$\Longleftrightarrow$(3b) and if $X$ is
normal, all properties are equivalent.

(1) $\mathsf{EC}(X,\mathbb{R})$ holds,

(2) $\mathsf{EC}(X,Y)$ holds when $Y$ is a metric space,

(3a) $X$ satisfies $\mathsf{I0}$,

(3b) given two non-Lindelöf $0$-sets of $X$, their intersection is non-
Lindelöf,

(4a) $X$ satisfies $\mathsf{IC}$,

(4b) given two closed non-Lindelöf subsets of $X$, their intersection is non-
Lindelöf.

###### Proof.

If $X$ is Lindelöf, we have nothing to do, hence we assume that $X$ is not
Lindelöf in what follows.
(2)$\Rightarrow$(1), (3b)$\Rightarrow$(3a), (4b)$\Rightarrow$(3b),
(4b)$\Rightarrow$(4a) are all immediate.
(1)$\Rightarrow$(3a). Let $f:X\to\mathbb{R}$ be given. If $A=f^{-1}(\\{0\\})$
is non-Lindelöf, then since $\mathsf{EC}(X,\mathbb{R})$ holds $f$ must be
eventually constant on $0$. Hence, since $f^{-1}(\mathbb{R}-\\{0\\})$ is a
countable union of $0$-sets, it is contained in a Lindelöf subset $Z$. Any
closed $B$ disjoint from $A$ is contained in $Z$ and hence Lindelöf.
(3a)$\Rightarrow$(2). Let $f:X\to Y$ be given, with $Y$ a metric space. Then
$f(X)$ has countable spread since any uncountable closed discrete subspace
$D\subset f(X)$ can be partitioned in two disjoint such subspaces whose
preimages yield disjoint non-Lindelöf $0$-sets of $X$. By Lemma 1.3 $f(X)$ is
hereditarily Lindelöf. Let $B(y,\epsilon)$ denote the open ball of radius
$\epsilon$ around $y$ in $f(X)$. For each $\epsilon>0$, by Lindelöfness there
is at least one $y\in f(X)$ such that $f^{-1}(B(y,\epsilon))$ is non-Lindelöf.
Moreover, if $f^{-1}(\overline{B(y,\epsilon)})$ is non-Lindelöf, then
$f^{-1}(f(X)-\overline{B(y,\epsilon)})$ is Lindelöf, because it is a countable
union of $0$-sets disjoint from $f^{-1}(\overline{B(y,\epsilon)})$. For each
$n\in\omega$, choose $y_{n}\in f(X)$ such that
$f^{-1}(\overline{B(y_{n},1/n)})$ is non-Lindelöf and $y_{n+1}\in
B(y_{n},1/n)$. Then $f(X)\bigcap\cap_{n\in\omega}\overline{B(y_{n},1/n)}$ is
non-empty (otherwise $f(X)=\cup_{n\in\omega}(f(X)-\overline{B(y_{n},1/n)})$
would have Lindelöf preimage) and thus contains exactly one point $y$. By
construction, the complement of $y$ has a Lindelöf preimage, hence $f$ is
eventually constant on $y$.
(4a)$\Rightarrow$(4b) Let $C_{1},C_{2}\subset X$ be closed and non-Lindelöf.
By way of contradiction assume that $C_{1}\cap C_{2}$ is (at most) Lindelöf.
By Lemma 3.1 (a), there are open $U,V$ both containing $C_{1}\cap C_{2}$ such
that $C_{1}-U$ and $C_{2}-V$ are disjoint non-Lindelöf closed subsets of $X$,
contradicting $\mathsf{IC}$.
(3a)$\Rightarrow$(3b) when $X$ is Tychonoff. Same proof as
(4a)$\Rightarrow$(4b), using Lemma 3.1 (b).
(3a)$\Rightarrow$(4a) if $X$ is normal. Given two disjoint closed sets $E,F$
in $X$ we obtain two disjoint $0$-sets $A\supset E,B\supset F$ with a Urysohn
fonction. If both $E$ and $F$ are non-Lindelöf, then so are $A$ and $B$,
contradicting (b). ∎

Half of next result is a direct consequence of Theorem 3.3.

######  Theorem 3.4.

If $\mathsf{EC}(X,\mathbb{R})$ holds and $X$ is either normal or
$\aleph_{1}$-scwH and Tychonoff, then $X$ is $\omega_{1}$-compact.

Recall that a space is $\kappa$-(strongly) collectionwise Hausdorff,
abbreviated $\kappa$-(s)cwH, iff for any closed discrete subspace $D$ of
cardinality $\leq\kappa$ there is a disjoint (resp. discrete) collection of
open sets $\mathcal{U}=\\{U_{d}\,:\,d\in D\\}$ such that $d\in U_{d}$ for each
$d\in D$. Such a $\mathcal{U}$ is called a disjoint (resp. discrete) expansion
of $D$. A normal $\kappa$-cwH space is $\kappa$-scwH, as well known.

###### Proof.

A closed discrete subset $D$ (of cardinality $\aleph_{1}$) can be partitioned
in two such subsets $D_{1},D_{2}$, which contradicts $\mathsf{IC}$. This
proves the result if $X$ is normal. If $X$ is Tychonoff and $\aleph_{1}$-scwH,
then $D$ is contained in a $0$-set $E$ which is the union of a discrete family
$\\{E_{d}\,:\,d\in D\\}$, where $d\in E_{d}$ for each $d\in D$. Indeed, take a
discrete expansion $\mathcal{U}=\\{U_{d}\,:\,d\in D\\}$ of $D$ and
$g_{d}:X\to[0,1]$ which is $1$ on $d$ and $0$ outside of $U_{d}$. By
discreteness, $g_{i}=\sum_{d\in D_{i}}g_{d}$ ($i=1,2$) is a continuous
function, and $E_{i}=g_{i}^{-1}(\\{1\\})$ ($i=1,2$) contradict $\mathsf{I0}$.
∎

Notice that there are non-$\omega_{1}$-compact spaces $X$ such that
$\mathsf{EC}(X,\mathbb{R})$ holds; Example 2.7 is one, Example 3.5 below is
another.

######  Example 3.5 (Nyikos, in effect).

There is a non-$\omega_{1}$-compact and non-normal Type I surface $M$ such
that $\mathsf{EC_{cl}}(M,\mathbb{R})$ and $\mathsf{S}(M,\mathbb{R})$ hold.

###### Details.

This example is due to Nyikos and is described in [24, section 6 & 7]. Every
property we will claim in this proof is proved in Nyikos’ paper to which we
refer for details. Let us give a quick description. We first consider a
tangent bundle $T\mathbb{L}_{+}$ of $\mathbb{L}_{+}$ given by a smoothing. The
removal the $0$-section $L_{0}$ of $T\mathbb{L}_{+}$, which is a copy of
$\mathbb{L}_{+}$, disconnects $T\mathbb{L}_{+}$ into two homeomorphic
connected submanifolds $T^{+}$ and $T^{-}$. Since $T\mathbb{L}_{+}$ is not
trivial, $T^{+}$ does not contain a copy of $\mathbb{L}_{+}$, actually there
is no unbounded map $\mathbb{L}_{+}\to T^{+}$. Write
$\pi:T\mathbb{L}_{+}\to\mathbb{L}_{+}$ for the bundle projection. Both
$T\mathbb{L}_{+}$ and $T^{+}$ are Type I manifolds, a canonical cover being
given by the fibers
$\\{U_{\alpha}=\pi^{-1}(0,\alpha)\,:\,\alpha\in\omega_{1}\\}$ (and their
intersections with $T^{+}$ for this latter space). for $\alpha\in\omega_{1}$.
The choice of the smoothing is important since $T\mathbb{L}_{+}$ and $T^{+}$
can exhibit quite different topological properties depending on it. Here, as
we need $\mathsf{EC_{cl}}(T^{+},\mathbb{R})$ (and thus
$\mathsf{EC_{cl}}(T^{-},\mathbb{R})$) to hold, we take what Nyikos calls a
smoothing of class 7, see [24, Section 6 & 7] for the construction and the
proof. Notice that $T^{+}$ is not normal. Set $M$ to be
$T\mathbb{L}_{+}-\omega_{1}$, where $\omega_{1}$ is seen as a subset of
$L_{0}$. Then $M$ is neither $\omega_{1}$-compact nor normal but
$\mathsf{EC_{cl}}(M,\mathbb{R})$ holds. Indeed, any $f:S\to\mathbb{R}$ is
eventually constant on both $T^{+}$ and $T^{-}$, by density and connectedness
it is eventually constant on all of $M$.
By construction, $U_{\alpha}$ is homeomorphic to $(0,\alpha)\times\mathbb{R}$
for each $\alpha$. Given $\alpha$, one may easily define
$r_{\alpha}:T\mathbb{L}_{+}\to U_{\alpha+1}$ which is the identity on
$\overline{U_{\alpha}}$ and sends all of $T\mathbb{L}_{+}-U_{\alpha+1}$ to a
point in $U_{\alpha+1}-\overline{U_{\alpha}}$, as seen in Figure 1. Then
$r_{\alpha}$ is well defined on $M$ for each $\alpha$. Together with
$\mathsf{EC_{cl}}(M,\mathbb{R})$, it implies that $\mathsf{S}(M,\mathbb{R})$
holds as well. ∎

Figure 1: The retraction $r_{\alpha}$ of Example 3.5. It is the identity on
the blue region.

We now investigate for which spaces $X$, $Y$ does $\mathsf{EC}(X,\mathbb{R})$
imply $\mathsf{EC}(X,Y)$. It might be the right time for a trivial remark: if
$Y\supset X$ and $X$ is not Lindelöf, then the inclusion $i:X\to Y$ shows that
$\mathsf{EC}(X,Y)$ does not hold. Since any Tychonoff space is a subspace of a
compact space (namely: its Čech-Stone compactification), it would be silly to
try to show that $\mathsf{EC}(X,Y)$ holds for some class of non-Lindelöf
Tychonoff spaces $X$ by relying only on covering properties of $Y$, we also
need some local properties. We shall see that having $G_{\delta}$ points plays
a central role.

A space is C-closed iff any countably compact subspace is closed. The class of
C-closed spaces contains in particular the hereditarily meta-Lindelöf spaces,
the sequential spaces and the regular spaces with $G_{\delta}$ points, see
e.g. [14]. The following lemma is immediate, although it implies for instance
[11, Lemma A.33].

######  Lemma 3.6.

Let $X$ be countably compact and $Y$ be C-closed. Then any $f:X\to Y$ is a
closed map.

Theorem 3.8 below is a slight generalization of [11, Lemmas B.29–30], with a
simpler proof. We first state a useful technical lemma from [5, Lemma 2.2].

Given a chain cover $\mathcal{U}=\\{U_{\alpha}\,:\,\alpha\in\kappa\\}$ of $X$,
a subset of $X$ is $\mathcal{U}$-unbounded iff it is not contained in any
$U_{\alpha}$. It is well known that a space is compact iff any chain cover is
trivial, that is, has a maximal element which is the whole space. (To our
knowledge, this is due to Alexandroff and Urysohn [1] in 1929.)

######  Lemma 3.7 ( [5, Lemma 2.2]).

Let $f:X\to Y$ be such that $X$ is countably compact and non-compact,
$f(X)\subset Y$ is compact, and $Y$ is C-closed. Let
$\mathcal{U}=\\{U_{\alpha}\,:\,\alpha\in\kappa\\}$, $\kappa\geq\omega_{1}$, be
a chain cover of $X$. Then, there is $c\in Y$ such that $f^{-1}(\\{c\\})$ is
$\mathcal{U}$-unbounded (and hence non-Lindelöf).

We include a proof for completeness.

###### Proof.

For each $\alpha\in\kappa$ choose $x_{\alpha}\in X-U_{\alpha}$ and set
$E_{\alpha}=\\{x_{\beta}\,:\,\beta\geq\alpha\\}$. Then
$\overline{E_{\alpha}}\cap U_{\alpha}=\varnothing$ and each $E_{\alpha}$ is
$\mathcal{U}$-unbounded. Since $\overline{E_{\alpha}}$ is countably compact,
$f(X)$ is compact and $Y$ is C-closed, it follows that
$f(\overline{E_{\alpha}})\subset f(X)$ is compact. Any finite intersection of
$E_{\alpha}$ is nonempty, hence by compactness
$\cap_{\alpha\in\kappa}f(\overline{E_{\alpha}})\not=\varnothing$. Take $c$ is
this intersection, then
$f^{-1}(\\{c\\})\cap\overline{E_{\alpha}}\not=\varnothing$ for each $\alpha$,
which shows that $f^{-1}(\\{c\\})$ is $\mathcal{U}$-unbounded. ∎

We state (and prove) the next theorem in an almost asurd level of generality.
The linear Lindelöf number $\ell L(X)$ of a space $X$ is the smallest $\kappa$
such that any chain cover of $X$ has a subcover of cardinality $\leq\kappa$. A
space is $[\aleph_{0},\kappa]$-compact iff any cover by at most $\kappa$ open
sets has a countable subcover.

######  Theorem 3.8.

Let $X$ be a countably compact non-compact space with linear Lindelöf number
$\kappa\geq\omega_{1}$. Let $Y$ be a $[\aleph_{0},\kappa]$-compact C-closed
space. Then the following holds.
(a) For each $f:X\to Y$, $f(X)$ is compact and there is $c\in Y$ with a non-
Lindelöf preimage in $X$.
(b) If $X$ satisfies $\mathsf{IC}$ and $Y$ has $G_{\delta}$ points, then
$\mathsf{EC}(X,Y)$ holds.
(c) If $X$ satisfies $\mathsf{I0}$ and $Y$ is Tychonoff with $G_{\delta}$
points, then $\mathsf{EC}(X,Y)$ holds.

This theorem is very similar to Lemma 6.11 in [5]. We will however repeat the
proof for completeness. Recall that as said earlier a regular space with
$G_{\delta}$ points is C-closed, hence in (c) there is some redundancy in the
assumptions. Actually, the image of $X$ in $Y$ is first countable in (b) and
(c) since a countably compact space with $G_{\delta}$ points is first
countable. (This actually holds even for feebly compact spaces, see [27, Prop.
2.2] for more on the subject.) Typical examples of $X$ satisfying the
hypothesis of (b) are $\omega_{1}$ and $\mathbb{L}_{\geq 0}$. Notice that if
$X$ is Type I, then $\kappa=\omega_{1}$, hence $Y$ may be assumed to be
$[\aleph_{0},\aleph_{1}]$-compact, which is equivalent to the property that
for any uncountable subset $B\subset Y$ there is a point of $Y$ any of whose
neighborhoods contains uncountably many members of $B$ – such a point is
called a condensation point of $B$. An immediate corollary of Theorems 3.3 &
3.8 is the following.

######  Corollary 3.9.

Let $X$ be countably compact and $Y$ be a Tychonoff Lindelöf space with
$G_{\delta}$ points. Then $\mathsf{EC}(X,\mathbb{R})$ is equivalent to
$\mathsf{EC}(X,Y)$.

###### Proof.

$Y$ is $[\aleph_{0},\kappa]$-compact for each $\kappa$ by Lindelöfness, and if
$\mathsf{EC}(X,\mathbb{R})$ holds then by Theorem 3.3 $X$ satisfies
$\mathsf{I0}$. Then $\mathsf{EC}(X,Y)$ follows by Theorem 3.8 (c). ∎

###### Proof of Theorem 3.8.

(a) Clearly, $f(X)$ is countably compact and closed in $Y$ and $\ell
L(f(X))\leq\ell L(X)=\kappa$. Given a chain cover $\mathcal{U}$ of $f(X)$,
there is a subcover of cardinality $\leq\kappa$. By
$[\aleph_{0},\kappa]$-compactness of $Y$, there is a countable subcover and
hence a finite one, so $f(X)$ is compact. Lemma 3.7 yields the result.
(b) Given $f:X\to Y$, by (a) there is $c\in Y$ with a non-Lindelöf preimage.
Let $V_{n}\subset Y$ be open sets such that $\\{c\\}=\cap_{n\in\omega}V_{n}$.
By $\mathsf{IC}$, $f^{-1}(Y-V_{n})$ must be Lindelöf, and hence
$f^{-1}(Y-\\{c\\})=\cup_{n\in\omega}f^{-1}(Y-V_{n})$ is also Lindelöf.
(c) Define $V_{n}$ as in (b) and take $g:Y\to\mathbb{R}$ which is $0$ on $c$
and $1$ on $Y-V_{n}$. Since $(g\circ f)^{-1}(\\{0\\})\supset f^{-1}(\\{c\\})$
is non-Lindelöf, $(g\circ f)^{-1}(\\{1\\})\supset f^{-1}(Y-V_{n})$ must be
Lindelöf, and we conclude as in (b). ∎

A quick corollary of Theorem 3.8 is the following.

######  Corollary 3.10.

Let $X$ be a countably compact space satisfying $\mathsf{IC}$. Then
$\mathsf{EC}(X,Y)$ holds whenever $Y$ is a C-closed isocompact space with
$G_{\delta}$ points.

A space is isocompact iff every closed countably compact subset is compact.
Hence a space is C-closed and isocompact iff each countably compact subset is
compact. Such a space is sometimes called hereditarily isocompact. For more on
isocompact spaces, see for instance [2, 7]. Note for instance that a meta-
Lindelöf space is isocompact.

###### Proof of Corollary 3.10.

Let $f:X\to Y$. Since $Y$ is C-closed, $Z=f(X)$ is a closed countably compact
subspace of $Y$. Thus $Z$ is compact, and in particular
$[\aleph_{0},\kappa]$-compact. We may thus apply (b) of Theorem 3.8. ∎

A particular case of Corollary 3.10 is the following:

######  Corollary 3.11 (A. Mardani, Prop. 4.3.12 and 4.3.23 in [17]).

Let $Y$ be either a realcompact space with $G_{\delta}$ points or a discrete
space. Then $\mathsf{EC}(\omega_{1},Y)$ holds.

###### Proof.

If $Y$ is discrete, each countably compact subspace is finite and hence
closed. Recall that a realcompact space is Tychonoff and hence C-closed if it
has $G_{\delta}$ points. A countably compact closed subset of a realcompact
space is compact, see e.g. [10, 3.11.1 & 3.11.4]. It follows that a
realcompact space is isocompact. ∎

Let us now show that if $Y$ is confined to regular spaces, then there is a way
to obtain (b) (and (a)) of Theorem 3.8 without the assumption that $X$ is
countably compact whenever $X$ is ‘narrow’ enough. The proof of the next
theorem is essentilally done in the preprint [5, Lemmas 6.10–6.11], but again,
we repeat it for completeness.

######  Theorem 3.12.

Let $X$ be a space such that any countable family of closed non-Lindelöf
subsets of $X$ has a non-Lindelöf intersection. Let $Y$ be a regular
$[\aleph_{0},\kappa]$-compact space with $G_{\delta}$ points, where
$\kappa=\ell L(X)$. Then $\mathsf{EC}(X,Y)$ holds.

A typical example for $X$ is a stationary subset of $\omega_{1}$ with subspace
topology.

###### Proof.

If $X$ is Lindelöf, there is nothing to do, hence we assume $X$ to be non-
Lindelöf. By assumption $X$ satisfies $\mathsf{IC}$. The proof of (b) in
Theorem 3.8 above does not use countable compactness and only relies on
$\mathsf{IC}$, the fact that points are $G_{\delta}$ and that there is $c\in
Y$ with non-Lindelöf preimage. We now show that the latter holds in our case
as well. Let thus $\mathcal{U}=\\{U_{\alpha}\,:\,\alpha\in\lambda\\}$
($\omega_{1}\leq\lambda\leq\kappa$) be a chain cover of $X$ such that
$X\not\subset U_{\alpha}$ for each $\alpha$, with $\lambda$ a regular
cardinal. Take $x_{\alpha}$ in $X-U_{\alpha}$ and set
$A_{\alpha}=\\{x_{\beta}\,:\,\beta>\alpha\\}$. Then $\overline{A_{\alpha}}\cap
U_{\alpha}=\varnothing$, each $\overline{A_{\alpha}}$ is
$\mathcal{U}$-unbounded, and a countable intersection of
$\overline{A_{\alpha}}$’s is nonempty. If $|f(A_{0})|<\lambda$, there is some
$c$ such that $f^{-1}(\\{c\\})$ is $\mathcal{U}$-unbounded. Else, let $c$ be a
point of $f(A_{0})$ such that any neighborhood of $c$ contains $\lambda$-many
points of $f(A_{0})$ (which exists by $[\aleph_{0},\kappa]$-compactness).
Since $Y$ is regular and points are $G_{\delta}$, there are open $U_{n}\subset
Y$, $n\in\omega$, such that $\\{c\\}=\cap_{n\in\omega}\overline{U_{n}}$. Then
$f^{-1}(\overline{U_{n}})$ is $\mathcal{U}$-unbounded (and thus non-Lindelöf)
in $X$ for each $n$. Hence, their intersection is non-Lindelöf as well and
equal to $f^{-1}(\\{c\\})$. ∎

It seems difficult to significantly weaken the hypothesis about $G_{\delta}$
points in Theorem 3.8, as the simple next example shows.

######  Example 3.13.

The quotient space $Y=\omega_{1}/\Lambda$ (where $\Lambda$ is the subspace of
limit ordinals) is Fréchet-Urysohn (hence $C$-closed), compact (hence
isocompact), but $\mathsf{EC}(\omega_{1},Y)$ does not hold, as shown by the
quotient map.

This example is also treated in [17, Ex. 4.2.42]. The only non-isolated point
of $Y$ is not a $G_{\delta}$ due to the pressing down lemma (see any book on
set theory, e.g. [16, Lemma 6.15]).

A canonical case in Corollary 3.10 is $X=\omega_{1}$. We now go the other way
and ask: Does $\mathsf{EC}(\omega_{1},Y)$, along with some ‘mild’ properties
of $Y$, imply isocompactness of $Y$ ? By a mild property we mean something
that is shared by many spaces but does not render the question trivial444We
are aware that this phrase is very similar to “By something yellow we mean
something containing yellowness”, which does not bring much enlightment.. An
interesting case is that of locally compact countably tight spaces, for which
the answer depends on the axioms of set theory. Firstly, Theorem 3.15 just
below shows that a ‘yes’ is consistent. It is yet another example of the
“ubiquity” of $\omega_{1}$ in countably compact non-compact spaces under the
proper forcing axiom PFA, and is actually only a restatement of other authors
old results from our point of view. Secondly, Example 3.16 below (also due to
another author) shows that ‘no’ is also consistent. We first show a small
lemma.

######  Lemma 3.14.

If $X=\cup_{\alpha\in\omega_{1}}X_{\alpha}$ is of Type I and countably
compact, then $X_{\alpha}\not=\overline{X_{\alpha}}$ for limit $\alpha$, so
$\cup_{\alpha\in\Lambda}(\overline{X_{\alpha}}-X_{\alpha})$ is a perfect
preimage of $\omega_{1}$.

Recall that a map is perfect iff it is closed and points have compact
preimages and that $\Lambda\subset\omega_{1}$ is the subset of limit ordinals.

###### Proof.

Take a strictly increasing sequence $\alpha_{n}$ whose limit is $\alpha$ and
for each $n$ some $x_{n}\in X_{\alpha_{n+1}}-X_{\alpha_{n}}$. This sequence
cannot have an accumulation point in $X_{\alpha}$. This shows that
$X_{\alpha}$ is not countably compact when $\alpha$ is limit. If $X$ itself is
countably compact, then $X_{\alpha}$ is not closed for limit $\alpha$, hence
$\overline{X_{\alpha}}-X_{\alpha}\not=\varnothing$. The map
$\cup_{\alpha\in\Lambda}(\overline{X_{\alpha}}-X_{\alpha})\to\omega_{1}$
sending each point in $\overline{X_{\alpha}}-X_{\alpha}$ to $\alpha$ is
perfect and its image is homeomorphic to $\omega_{1}$. ∎

######  Theorem 3.15 (PFA).

Let $Y$ be a countably tight space such that $\mathsf{EC}(\omega_{1},Y)$
holds. If $Y$ is moreover either of Type I or locally compact, then $Y$ is
isocompact.

###### Proof.

We assume first that $Y=\cup_{\alpha\in\omega_{1}}Y_{\alpha}$ is of Type I.
Suppose that $Y$ is not isocompact, then $Y$ has a countably compact closed
subset $Z$ which is not compact. Since each $\overline{Y_{\alpha}}$ is
Lindelöf, $Z\not\subset\overline{Y_{\alpha}}$ so $Z$ is actually a Type I
countably compact space with $Z_{\alpha}=Z\cap Y_{\alpha}$. By Lemma 3.14
$\cup_{\alpha\in\Lambda}(\overline{Z_{\alpha}}-Z_{\alpha})$ is a perfect
preimage of $\omega_{1}$ which is moreover countably tight. (The inclusion
$\overline{Z_{\alpha}}\subset Z\cap\overline{Y_{\alpha}}$ could be strict, but
$\overline{Z_{\alpha}}$ is compact anyway.) By [9] it contains a copy of
$\omega_{1}$ under PFA, and there is a non-eventually constant map
$\omega_{1}\to Y$.
Assume now that $Y$ is locally compact. A closed subset of $Y$ is also locally
compact. By [6, Thm 2.6], under PFA a locally compact countably compact non-
compact space contains a perfect preimage of $\omega_{1}$, and we conclude as
above. ∎

The full force of PFA is probably not needed in the proof, but it seems that
more than MA+$\neg$CH is, as the following examples (both due to Nyikos in
[19]) show. The axiom $\clubsuit_{C}$ is a weakening of $\diamondsuit$
compatible with MA+$\neg$CH. See Nyikos article for details. (It is worth
noting that [19] is still a preliminary draft at the time of writing, but
there is a similar construction of a surface under $\diamondsuit$ in [20, Ex.
6.17], and we did not find anything dubious in Nyikos’ construction.)

######  Example 3.16 ($\clubsuit_{C}$).

There is a a longpipe (thus $\omega$-bounded $2$-manifold) $Y$ which is not
isocompact but $\mathsf{EC}(\omega_{1},Y)$ holds. Also, there is a $2-1$
closed preimage of $\omega_{1}$ $P$ with the same properties.

###### Details.

The surface is described in [19, Section 5] and the so called ‘sprat’ $P$ is
given by Theorem 2.1 in the same paper and the remarks after. Both are regular
and first countable and hence C-closed, Type I, countably compact, non-
compact, and neither contain a copy of $\omega_{1}$. Actually, both satisfy a
property stronger than (c) of Theorem 3.3: any club subset of $Y$ (resp. of
$P$) contains $\overline{Y_{\alpha}}-Y_{\alpha}$ (resp.
$\overline{P_{\alpha}}-P_{\alpha}$) for a club set of $\alpha$. Moreover any
bounded subset of $Y$ or $P$ embeds in $\mathbb{R}^{2}$. (This is a detail,
but in Nyikos construction $Y_{\alpha}$ is homeomorphic to $\mathbb{R}^{2}$,
however choosing a small closed disk in $Y_{0}$ and removing its interior, we
obtain $Y_{\alpha}$ homeomorphic to the cylinder, as in our definition of
longpipe.)

The next lemma shows that a map $f$ from $\omega_{1}$ to $Y$ or $P$ must have
a bounded image, and since any bounded subset of $Y$ or $P$ embeds in
$\mathbb{R}^{2}$, $f$ must be eventually constant. ∎

######  Lemma 3.17.

If $Y$ is a Type I C-closed space and $f:\omega_{1}\to Y$ is unbounded, then
there is a copy of $\omega_{1}$ in $Y$.

###### Proof.

The image of $f$ is club by Lemma 3.6, hence by a routine argument similar to
that of [11, Lemma 1.19], the set
$C=\\{\alpha\in\omega_{1}\,:\,f(\alpha)\in\overline{Y_{\alpha}}-Y_{\alpha}\\}$
is club as well. Thus $C$ embeds in $Y$ and is a copy of $\omega_{1}$. ∎

We note that there are longpipes $Y$ (which are of course non isocompact)
satisfying $\mathsf{EC}(\mathbb{L}_{\geq 0},Y)$ in ZFC, see Example 7.7 below.

##  4 $\mathsf{S}(T,\mathbb{R})$ and Suslin trees

A tree $T$ is a partially ordered set such that each point has a well ordered
set of predecessors. We usually denote the order by $<$, and $>,\leq,\geq$ are
defined as usual. Here all trees are endowed with the order topology (also
called interval topology): a basis is given by the intervals $\\{z\in
T\,:\,x<z\leq y\\}$ for each $x,y\in T$. We assume that our trees are
Hausdorff, that is, if $x,y\in T$ are at a limit level and have the same
predecessors, then $x=y$. It follows that any tree is a $0$-dimensional space.
A chain is a totally ordered subset and an antichain a subset with pairwise
incomparable elements. The $\alpha$-th level of $T$ consists of the members
whose set of predecessors has order type $\alpha$. Points in the $\alpha$-th
level are often said to be at height $\alpha$. The height of $T$ is the
smallest ordinal $\beta$ such that the $\beta$-th level of $T$ is empty.
An $\omega_{1}$-tree has countable levels and height $\omega_{1}$. A tree is
Suslin if it has height $\omega_{1}$ and its chains and antichains are at most
countable. Recall that Suslin trees do not exist in ZFC alone, but do exist
under V=L or $\diamondsuit$, for instance. A subset $D\subset T$ is order
dense iff given any $x\in T$ there is $y\in D$ with $y>x$. For a tree $T$ and
$t\in T$, write $T_{\geq t}=\\{s\in T\,:\,s\geq t\\}$,
$t\upharpoonright\alpha$ for the unique predecessor of $t$ at level $\alpha$
(if $t$ is below the $\alpha$-th level, $t\upharpoonright\alpha=t$) and
$T_{<\alpha}$ for the subset of elements at level $<\alpha$. If $T$ is an
$\omega_{1}$-tree then it is a Type I space and the $T_{<\alpha}$ form a
canonical cover, moreover each $T_{<\alpha}$ embeds in $\mathbb{R}$. (For the
latter claim, notice that $T_{<\alpha}$ is a secound countable $0$-dimensional
space, hence embeds in the Cantor set.)
Given an ordinal $\alpha$, define $r_{\alpha}:T\to T_{<\alpha+1}$ as
$r_{\alpha}(t)=t\upharpoonright\alpha$. Notice that if $\beta\geq\alpha$ and
$f\circ r_{\alpha}=f$ for some $f:T\to Y$, then $f\circ r_{\beta}=f$ as well.
If $A\subset T$, denote by $A^{\downarrow}$ its downward closure $\\{x\in
T\,:\,\exists y\in A\;x\leq y\\}$. If $x\in A\subset T$ we let $A_{\geq x}$ be
$A\cap T_{\geq x}$. Since we are going to look at $\omega_{1}$-compact
subspaces of trees, let us recall the following (more or less classical)
facts.

######  Lemma 4.1.

Let $T$ be a tree of height $\omega_{1}$ and $S\subset T$ be uncountable and
$\omega_{1}$-compact in the subspace topology. Then the following hold.
(a) A closed discrete subset of $T$ is a countable union of antichains, and an
antichain of $T$ is closed discrete.
(b) If $A\subset T$, $A$ has an uncountable antichain if and only if
$A^{\downarrow}$ has one.
(c) $S^{\downarrow}$ is the union of a countable set, at most countably many
copies of $\omega_{1}$ and a Suslin tree.
(d) There is $\alpha\in\omega_{1}$ such that $|(S^{\downarrow})_{\geq
x}|=|S_{\geq x}|\geq\aleph_{1}$ when $x$ is above level $\alpha$ in
$S^{\downarrow}$.
(e) $S$ intersects a stationary subset of levels of $S^{\downarrow}$.
Moreover, any closed subset of $S^{\downarrow}$ intersects a stationary subset
of levels.
(f) Let $E,F\subset S$ be closed (in $S$). If $|E\cap F|\leq\aleph_{0}$, then
$|E^{\downarrow}\cap F^{\downarrow}|\leq\aleph_{0}$.
(g) If $A\subset S^{\downarrow}$ is uncountable, there is $x\in
S^{\downarrow}$ such that $(S^{\downarrow})_{\geq x}\subset A^{\downarrow}$.
(h) If $D\subset S^{\downarrow}$ is order-dense and upward-closed, then
$D\supset S^{\downarrow}-(S^{\downarrow})_{\beta}$ for some $\beta$.
(i) If $U\subset S$ is open in $S$ such that $U$ intersects stationary many
levels of $(S^{\downarrow})_{\geq x}$ for each $x\in S$, then
$U\supset(S^{\downarrow}-(S^{\downarrow})_{\beta})\cap S$ for some $\beta$.

###### Proof.

All are probably part of the folklore, actually they are well known facts when
$T=S$ is a Suslin tree. Proofs of items (b) to (g) can be found in the
preprint [5, Lemmas 6.6–6.7], and (a) in [25, Theorem 4.11], for instance. By
considering minimal elements, it is easy to see that (h) holds if
$S^{\downarrow}$ is Suslin, hence it follows by (c). Let us prove (i). Suppose
that $U$ is as in the statement of (i). Let $\alpha$ be given by (d). By
removing the $\alpha$-th first levels, we may assume that $S_{\geq x}$ is
uncountable for all $x\in S$. By (h) it is enough to show that $F=\\{x\in
S\,:\,S_{\geq x}\subset U\\}$ is order-dense in $S$ (or equivalently in
$S^{\downarrow}$) above level $\alpha$. Fix $z\in S$ and let $W$ be open in
$S^{\downarrow}$ such that $W\cap S=U$. Then $W$ intersects stationary many
limit levels of $(S^{\downarrow})_{\geq z}$. For each $x\in W$ at a limit
level with $x>z$, let $\sigma(x)<x$ be such that $\\{u\,:\,\sigma(x)\leq u\leq
x\\}\subset W$. By the pressing-down lemma for $\omega_{1}$-trees (see, e.g.,
[12, p. 154]) there is some $y\in S^{\downarrow}$ such that
$E=\sigma^{-1}(\\{y\\})$ meets stationary many levels of
$(S^{\downarrow})_{\geq z}$. For each $x\in E$ the segment between $y$ and $x$
is contained in $W$. We may assume that $y\geq z$. By (g), there is some
$x\geq y$ such that $(S^{\downarrow})_{\geq x}\subset E^{\downarrow}$. But
this means that $(S^{\downarrow})_{\geq x}\subset W$, hence $S_{\geq x}\subset
U$. We thus proved that there is a point $x$ of $F$ above $z$, and hence $F$
is order-dense. ∎

The following was proved by Steprāns [31] when $S=T$. (His statement is
weaker, but he gives two proofs which actually show more than stated.) A proof
for $S\not=T$, adapted from Steprāns’, is given in [5, Lemma 6.7 (i)]. We
present another proof, by forcing, also based on Steprāns’ ideas.

######  Theorem 4.2.

Let $T$ be a tree of height $\omega_{1}$, $S\subset T$ be uncountable and $Y$
be a space. Then the following hold.
(a) If $S$ is $\omega_{1}$-compact for the subspace topology and $Y$ is
submetrizable, then $\mathsf{S}(S,Y)$ holds. If $S=T$ then the retraction is
given by $r_{\alpha}$ for some $\alpha$.
(b) If $|Y|\geq 2$ is a space and $\mathsf{S}(S,Y)$ holds, then $S$ is
$\omega_{1}$-compact.

Recall that a space is submetrizable iff it has a coarser metrizable topology.

###### Proof.

We prove (b) first.
(b) Suppose that $S$ is not $\omega_{1}$-compact. By Lemma 4.1 (a)–(b) and
(d), $S$ contains an uncountable antichain $A$ consisting of isolated points
of $S$ and thus a clopen discrete uncountable subspace. Then, apply Lemma 2.15
(b).
(a) We may assume that $Y$ is metrizable by Corollary 2.13. By Lemma 4.1 (c),
the set $E$ containing the minimal points of

$\\{x\in S\,:\,S_{\geq x}\text{ is uncountable and totally ordered}\\}$

is at most countable. If $x\in E$, then $S_{\geq x}$ is (homeomorphic to) an
$\omega_{1}$-compact and hence stationary subset of $\omega_{1}$. Recall that
hereditary Lindelöfness and spread are equal in a metrizable space (see Lemma
1.3); hence, $f(S)$ is Lindelöf and by Theorem 3.12 $f$ is eventually constant
on $S_{\geq x}$ for each $x\in E$. We may thus assume by Lemma 4.1 (c) that
$S^{\downarrow}$ is a Suslin tree. We may take out the set of $x$ such that
$|f(S_{\geq x})|=1$. If what remains is countable, we are over, otherwise by
Lemma 4.1 (d) we may assume that for each $x\in S$, $S_{\geq x}$ is
uncountable and $|f(S_{\geq x})|>1$. We can also assume that $S^{\downarrow}$
is rooted by adding a common root below its minimal elements. We now use a
forcing argument, and force with $S^{\downarrow}$ with the reverse order. Let
$G$ be a generic filter. Recall that if $\langle X,\tau\rangle$ is a
topological space in the ground model, then $\tau$ serves as a base for the
topology $\tau(G)$ of $X$ on a forcing extension by $G$. Thus, any function
that is continuous in the ground model remains so in the forcing extension,
and $Y$ remains metrizable. Since $D_{\alpha}=\\{x\in
S^{\downarrow}\,:\,\text{height}(x)\geq\alpha\\}$ is dense, $G$ is an new
uncountable branch in $S^{\downarrow}$. Forcing with a Suslin tree preserves
stationarity and cardinals, actually it does not add countable sets to the
universe, see for instance [16, Exercices VII H1–H2 and Theorems VII.5.10 &
VII.8.4]. Hence, $\omega_{1}^{V}=\omega_{1}^{V[G]}$. Since $S$ intersects
stationary many levels of $S^{\downarrow}$ in $V$ by Lemma 4.1 (e) it does so
in $V[G]$ as well. Hence, $S$ intersects stationary many levels of $G$ and
$S\cap G$ is homeomorphic to a stationary subset of $\omega_{1}$. By Theorem
3.12, $f$ is constant on $G$ above some height; hence, there must be
$\alpha\in\omega_{1}$, $s\in S$, $u\in Y$ with
$s\Vdash\check{f}\left(\check{G}-\check{S}_{\check{\alpha}}\right)=\\{\check{u}\\}$.
But since $|f(S_{\geq x})|>1$ for any $x\in S$, $\\{z\in S\,:\,f(z)\not=u\\}$
is order-dense in $S^{\downarrow}$, and we have that
$1\Vdash\check{f}\left(\check{G}-\check{S}\upharpoonright\check{\alpha}\right)\not=\\{\check{u}\\}$,
a contradiction. It follows that $|f(S_{\geq x})|=1$ if $x$ is above some
level $\alpha$. Hence, given $y$ at level $>\alpha$, we may set
$r(y)=\min\\{x\leq y\,:\,\text{height}(x)\geq\alpha\\}$. If $S=T$, it is
enough to set $r(y)=y\upharpoonright\alpha$ (i.e. $r=r_{\alpha}$) since there
are points at every level. ∎

One consequence of Theorem 4.2 is the following.

######  Lemma 4.3.

Let $T$ be an $\omega_{1}$-compact tree of height $\omega_{1}$ (in particular
a Suslin tree), $Y$ be a space and assume that $\mathsf{S}(T,Y)$ holds. Then,
given $f:T\to Y$, the retraction witnessing $\mathsf{S}(T,Y)$ can be chosen to
be $r_{\alpha}$ for some $\alpha$.

###### Proof.

Let $f:T\to Y$ be given and $r:T\to Z\subset T_{\alpha+1}$ be a retraction
such that $f\circ r=f$. Since $T$ is $\omega_{1}$-compact, by Theorem 4.2
$\mathsf{S}(T,\mathbb{R})$ holds. Since $T_{\alpha+1}$ can be embedded in
$\mathbb{R}$, there is some $\alpha$ such that $r\circ r_{\alpha}=r$; hence,

$f\circ r_{\alpha}=(f\circ r)\circ r_{\alpha}=f\circ(r\circ r_{\alpha})=f\circ
r=f.$

∎

This has the following corollary.

######  Corollary 4.4.

Let $Y_{n}$, $n\in\omega$, be spaces and $T$ be a tree of height $\omega_{1}$.
Then $\mathsf{S}(T,Y_{n})$ holds for each $n\in\omega$ iff
$\mathsf{S}(T,\prod_{n\in\omega}Y_{n})$ holds.

###### Proof.

The reverse implication is immediate. Assume that $\mathsf{S}(T,Y_{n})$ holds
for each $n\in\omega$. Let $\pi_{n}$ be the projection on the $n$-th factor.
If $|Y_{n}|=1$ for each $n$, there is nothing to prove. Since
$\mathsf{S}(T,Y_{n})$ holds and $|Y_{n}|\geq 2$ for some $n$, by Theorem 4.2
(b) $T$ is $\omega_{1}$-compact. Let $f:T\to\prod_{n\in\omega}Y_{n}$. By Lemma
4.3, for each $n\in\omega$ there is $\alpha_{n}$ such that $\pi_{n}\circ
f\circ r_{\alpha_{n}}=\pi_{n}\circ f$. Hence $f\circ r_{\alpha}=f$ for
$\alpha=\sup_{n}\alpha_{n}$. ∎

Another simple consequence of Theorem 4.2 and Lemma 4.3 is that
$\mathsf{S}(T,Y)$ implies $\mathsf{BR}(T,Y)$ when $T$ is an $\omega_{1}$-tree.

######  Proposition 4.5.

Let $T$ be a tree of heigth $\omega_{1}$ and let $Y$ be any space. Then the
following holds.
(a) $\mathsf{S}(T,Y)\Rightarrow\mathsf{BR}(T,Y)$.
(b) If $|Y|\geq\aleph_{1}$, then
$\mathsf{S}(T,Y)\Leftrightarrow\mathsf{BR}(T,Y)\Leftrightarrow\mathsf{L}(T,Y)$.

###### Proof.

(a) If $|Y|=1$ there is nothing to prove, we assume $|Y|>1$. If
$\mathsf{S}(T,Y)$ holds, then by Theorem 4.2 (b) $T$ cannot have an
uncountable antichain; hence, $T$ is $\omega_{1}$-compact. By Lemma 4.1 (d),
there is $\gamma\in\omega_{1}$ such that $T_{\geq x}$ is uncountable if $x$ is
above level $\gamma$. Given $f:T\to Y$, by Lemma 4.3 there is some $\alpha$
such that $f\circ r_{\alpha}=f$. We can take $\alpha\geq\gamma$. Then, for
each $\beta\geq\alpha$,
$f(T-T_{<\beta})=f(T-T_{<\alpha})=f(\\{x\,:\,\text{height}(x)=\alpha\\})$.
This shows that $\mathsf{BR}(T,Y)$ holds.
(b) By (a) and Lemma 2.1, it is enough to show that
$\mathsf{L}(T,Y)\Rightarrow\mathsf{S}(T,Y)$. By Lemma 2.15, if
$|Y|\geq\aleph_{1}$ and $T$ has an uncountable antichain $A$, then
$\mathsf{L}(T,Y)$ does not hold. Hence, if $\mathsf{L}(T,Y)$ holds, then $T$
is $\omega_{1}$-compact. It follows that $\mathsf{S}(T,Y)$ holds by Theorem
4.2. ∎

Notice in passing that, in all generality, $\mathsf{S}$ and $\mathsf{BR}$ are
not equivalent for $\omega_{1}$-trees.

######  Example 4.6.

Let $T$ be a tree containing an uncountable antichain and $Y$ be a countable
space. Then $\mathsf{BR}(T,Y)$ holds but $\mathsf{S}(T,Y)$ does not.

(The property $\mathsf{BR}(T,Y)$ holds by Lemma 2.14.) When trying to
generalize Theorem 4.2 beyond the metrizable case, we are held back by the
following fact:

######  Example 4.7.

Let $T$ be a Suslin tree such that each element has infinitely many immediate
successors. Then, there is a a $1$-to-$1$ continuous $f:T\to Y$ where $Y$ is a
hereditarily Lindelöf first countable monotonically normal space.

###### Details.

$Y$ is actually the tree $T$ itself with the topology given by a
lexicographical ordering of $T$ chosen so that the resulting space is a Suslin
line (see just below). The lexicographical order $\leq_{\ell}$ in $T$ is
obtained by first considering a total order $\prec$ on $T$ and then letting
$y<_{\ell}x$ if and only if $y<x$ or $y\perp x$ and
$y\upharpoonright\alpha\prec x\upharpoonright\alpha$, where $\alpha$ is
minimal such that $y\upharpoonright\alpha\not=x\upharpoonright\alpha$. Denote
the topologies induced by $<,<_{\ell}$ on $T$ by $\tau_{<},\tau_{<_{\ell}}$.
The $<$-minimal elements of an $<_{\ell}$ interval $\\{y\in
T\,:\,x<_{\ell}y<_{\ell}z\\}$ cannot be at limit height because the tree is
Hausdorff. It follows that any such $<_{\ell}$-interval is a union of branches
starting at a successor level and is thus $\tau_{<}$-open in $T$. Hence
$id:\langle T,\tau_{<}\rangle\to\langle T,\tau_{<_{\ell}}\rangle$ is
continuous. It is well known that if $\prec$ orders the immediate successors
of any given point as $\mathbb{Q}$, then $\langle T,<_{\ell}\rangle$ is a
Suslin line, see for instance [15, Lemma 14.21] for a proof. We recall that a
Suslin line is a linearly ordered topological space which is ccc but not
separable. Every linearly ordered space is monotonically normal and every
Suslin line is first countable and hereditarily Lindelöf. Recall that a
regular hereditarily Lindelöf space is also perfect. ∎

This shows in particular that Theorems 3.8 and 3.12 with $\mathsf{S}$ instead
of $\mathsf{EC}$ do not hold for Suslin trees. But we can still say something
if the conclusion of Lemma 3.7 holds.

######  Lemma 4.8.

Let $Y$ be a regular space with $G_{\delta}$ points such that for each Suslin
tree $T$ and each $f:T\to Y$, there is $y\in Y$ with an uncountable preimage.
Then $\mathsf{S}(T,Y)$ holds for any Suslin tree $T$.

###### Proof.

Let $T$ be a Suslin tree. By Lemma 4.1 (d) we may assume that $T_{\geq x}$ is
uncountable for each $x\in T$. By Lemma 4.1 (h), it is enough to show that
$E=\\{x\in T\,:\,\exists y\in Y\text{ with }f(T_{\geq x})=\\{y\\}\\}$ is
order-dense in $T$. Let thus $z\in T$. Since $T_{\geq z}$ is Suslin there is
$y$ such that $A=f^{-1}(\\{y\\})$ is uncountable. Hence, by Lemma 4.1 (g)
there is $w\in T_{\geq z}$ such that $T_{\geq w}\subset A^{\downarrow}$. Since
$A$ is closed, by Lemma 4.1 (e) it intersects a stationary subset of levels of
$T_{\geq u}$ for each $u\geq w$. Let $U_{n}\subset Y$ be open such that
$\cap_{n\in\omega}U_{n}=\\{y\\}$. Then each $f^{-1}(U_{n})$ is open, order-
dense and intersects stationary many levels of $T_{\geq u}$ for each $u\geq
w$. By Lemma 4.1 (i), $A=f^{-1}(\\{y\\})=\cap_{n\in\omega}f^{-1}(U_{n})$
contains all of $T_{\geq w}$ above some level. This shows that $E$ is order-
dense. ∎

We do not know whether this lemma is of any use, that is, if there are spaces
satisfying its hypotheses but not those of Theorems 4.2.

##  5 $\mathsf{BR}(X,Y)$, $\mathsf{L}(X,Y)$ when $Y$ is metric and $X$ an
increasing chain of ‘nice’ subspaces

The goal of this section is to prove in particular the following theorem. We
recall that manifolds are assumed to be connected.

######  Theorem 5.1.

Let $M$ be a manifold. Then $\mathsf{L}(M,\mathbb{R})$ holds.

The proof relies on properties of increasing subsets of $\mathbb{R}$, and more
generally of metric spaces. We start by specifying some notation. Let
$\mathcal{P}$ be a topological property and $\gamma$ be an ordinal. We say
that a space $X$ is a $\mathcal{P}$-chain of length $\gamma$ iff
$X=\cup_{\alpha<\gamma}H_{\alpha}$ with $H_{\alpha}\subset H_{\beta}$ if
$\alpha<\beta$ and such that $H_{\alpha}$ has property $\mathcal{P}$ for each
$\alpha$. If $H_{\alpha}\subsetneq H_{\beta}$ for each $\alpha<\beta$, we say
that the chain is strict. If the property $\mathcal{P}$ cannot be described in
one word, we may use brackets. For instance, left separated spaces are strict
[closed-and-countable]-chains of length $\omega_{1}$. Type I spaces are both
strict [closed-and-Lindelöf]-chains and strict open-chains of length
$\omega_{1}$. A subspace $E\subset X$ such that $X-E$ satisfies $\mathcal{P}$
is said to be co-$\mathcal{P}$. We will use the following classical fact.

######  Lemma 5.2.

Let $Y$ be a metric space.
(a) $Y$ does not contain a strict compact-chain or a strict [co-compact]-chain
of length $\geq\omega_{1}$.
(b) If $Y$ is separable, then $Y$ does contain neither a strict open-chain nor
a strict closed-chain of length $\geq\omega_{1}$.

We provide a proof for completeness. Lemma 1.3 is used several times
implicitely.

###### Proof of Lemma 5.2.

We show (b) first.
(b) $Y$ is hereditarily Lindelöf and hereditarily separable. A strict open-
chain of length $\omega_{1}$ is not Lindelöf and a strict closed-chain of
length $\omega_{1}$ contains a non-separable subspace (see the proof of
Theorem 3.1 in [28]).
(a) If $\langle H_{\alpha}\,:\,\alpha<\omega_{1}\rangle$, is a compact-chain
of subspaces, then $H=\cup_{\alpha<\omega_{1}}H_{\alpha}$ is countably compact
and hence compact. Indeed, any countable subset of $H$ is contained in some
compact $H_{\alpha}$ and has thus an accumulation point. Hence $H$ is
separable, let $D$ be a countable dense subset. Then $D$ is contained in some
$H_{\alpha}$, by closedness $H=\overline{D}=H_{\alpha}$ and the chain cannot
be strict. If $\langle H_{\alpha}\,:\,\alpha<\omega_{1}\rangle$, is a chain of
co-compact subspaces, then $H_{\alpha}\cap W$ is open in $W=Y-H_{0}$ which is
compact and hence separable. By (b) the open chain $\langle H_{\alpha}\cap
W\,:\,\alpha<\omega_{1}\rangle$ in $W$ cannot be strict. Hence the chain of
$H_{\alpha}$ is not strict either. ∎

######  Corollary 5.3.

$\mathbb{R}$ does contain neither a strict connected-chain nor a strict [co-
connected]-chain of length $\omega_{1}$.

###### Proof.

Since the connected subsets of $\mathbb{R}$ are the intervals, an infinite
strict connected-chain or [co-connected]-chain contains a strict open-chain of
the same length. ∎

We say that $X$ is a slowpen chain if is a strict open-chain
$X=\cup_{\alpha\in\gamma}U_{\alpha}$, such that
$\cup_{\beta<\alpha}U_{\beta}=U_{\alpha}$ when $\alpha$ is limit and
$U_{\alpha+1}=U_{\alpha}\cup E_{\alpha}$, where $E_{\alpha}$ is Lindelöf for
each $\alpha$.

######  Lemma 5.4.

Let $X=\cup_{\alpha<\gamma}U_{\alpha}$ be a slowpen chain. If $U_{\alpha}$ is
connected for each $\alpha$, then $\mathsf{L}(X,\mathbb{R})$ holds.

###### Proof.

If $\gamma<\omega_{1}$, $X$ is Lindelöf and there is nothing to prove. Assume
that $\gamma\geq\omega_{1}$. Let $f:X\to\mathbb{R}$ be given. Let
$U_{\alpha},E_{\alpha}$, $\alpha<\gamma$ be as in the definition of a slowpen
chain, hence $X=\cup_{\alpha<\gamma}U_{\alpha}$. We prove by induction on
$\alpha$ that there is a Lindelöf subset $L_{\alpha}\subset X$ such that
$f(L_{\alpha})=f(U_{\alpha})$. This yields the theorem when $\alpha=\gamma+1$.
If $\alpha$ is countable, $U_{\alpha}$ is Lindelöf hence we may set
$L_{\alpha}=U_{\alpha}$. If $\alpha=\beta+1$, then the Lindelöf subset
$L_{\alpha}=L_{\beta}\cup E_{\beta}$ satisfies $f(U_{\alpha})=f(L_{\alpha})$.
If $\text{cof}(\alpha)=\omega$, choose an $\omega$-sequence
$\beta_{n}\nearrow\alpha$ and set $L_{\alpha}=\cup_{n\in\omega}L_{\beta_{n}}$.
Then $L_{\alpha}$ is Lindelöf and $f(U_{\alpha})=f(L_{\alpha})$. Assume now
that $\text{cof}(\alpha)>\omega$. Since $U_{\beta}$ is connected for each
$\beta<\alpha$, $f(U_{\beta})\subset\mathbb{R}$ is connected. Hence, $\langle
f(U_{\beta})\,:\,\beta<\alpha\rangle$ does not have a cofinal strict subchain
by Corollary 5.3. It follows that there is some $\beta<\alpha$ such that
$f(U_{\alpha})=f(U_{\beta})=f(L_{\beta})$. ∎

######  Lemma 5.5.

Let $M$ be a manifold. Then $M$ is a slowpen chain
$\cup_{\alpha<\gamma}U_{\alpha}$ of length $\gamma<\mathfrak{c}^{+}$ with each
$U_{\alpha}$ connected. Moreover, each $U_{\alpha}$ is a submanifold of $M$
and $M$ is metrizable iff $\gamma$ is countable.

###### Proof.

Our argument is very similar to the proof of Theorem 2.9 in [20] where it is
shown that the cardinality of a manifold is the continuum. Recall that
(hereditary) Lindelöfness and metrizability are equivalent for manifolds (see
e.g. [11, Thm 2.1]). Fix an Euclidean open connected open subset $U_{0}$. We
proceed by induction and assume that the chain of connected open sets
$\cup_{\beta<\alpha}U_{\beta}$ is a slowpen chain. If $\alpha$ is limit, we
let $U_{\alpha}=\cup_{\beta<\alpha}U_{\beta}$. Given $U_{\alpha}$, if
$U_{\alpha}=M$, we are over. If not, by connectedness of $M$ we have that
$\overline{U_{\alpha}}\not=U_{\alpha}$, so we may choose a point
$x\in\overline{U_{\alpha}}-U_{\alpha}$. We then let $U_{\alpha+1}$ be the
union of $U_{\alpha}$ and an Euclidean connected set $E_{\alpha}$ containing
$x$. Connectedness of $U_{\alpha}$ for each $\alpha$ is immediate. By
construction the chain is slowpen and strict up to $\alpha$. Since
$U_{\alpha}$ is open, it is a submanifold of $M$. Since $|M|=\mathfrak{c}$,
there is some $\gamma<\mathfrak{c}^{+}$ such that $M=U_{\gamma}$. If $\gamma$
is countable, $M$ is Lindelöf and hence metrizable. If $\gamma$ is
uncountable, it contains the non-Lindelöf subset
$\cup_{\alpha<\omega_{1}}U_{\alpha}$, which shows that $M$ is not metrizable.
∎

Theorem 5.1] follows immediately by Lemmas 5.4–5.5. Notice however that
$\mathbb{R}$ cannot be replaced by $\mathbb{R}^{2}$, as shown by Example 6.4
below. Also, we cannot replace $\mathsf{L}$ by $\mathsf{BR}$.

######  Example 5.6.

There is a surface (without boundary) $P$ such that
$\mathsf{BR}(P,\mathbb{R})$ does not hold.

###### Details.

This example is classical: Take $P$ to be the Prüfer surface (separable
version with boudary)
$\mathbb{R}\times\mathbb{R}_{>0}\cup\displaystyle\cup_{a\in\mathbb{R}}\mathbb{R}_{a}$,
where each $\mathbb{R}_{a}$ is a (distinct) copy of $\mathbb{R}$. See [11, Ex.
1.25] for a complete description of the topology. This surface can be seen as
taking $\mathbb{R}\times\mathbb{R}_{\geq 0}$ and ‘blowing up’ each point
$\langle a,0\rangle$ in the bottom boundary into the open interval
$\mathbb{R}_{a}$. This gives again a surface whose boundary components are the
$\mathbb{R}_{a}$. If $A\subset\mathbb{R}$, then
$\mathbb{R}\times\mathbb{R}_{>0}\cup\displaystyle\cup_{a\in A}\mathbb{R}_{a}$
is an open submanifold of $P$, hence a Lindelöf subset of $P$ intersects at
most countably many $\mathbb{R}_{a}$. Any continuous
$f:\mathbb{R}\times\mathbb{R}_{\geq 0}\to\mathbb{R}$ yields a continuous
$f_{P}:P\to\mathbb{R}$ by letting $f_{P}$ be constant with value $f(a)$ on
$\mathbb{R}_{a}$ and equal to $f$ on $\mathbb{R}\times\mathbb{R}_{>0}$. Let
$f:\mathbb{R}\times\mathbb{R}_{\geq 0}\to\mathbb{R}$ be the projection on the
first coordinate. Then $f_{P}$ shows that $\mathsf{BR}(P,\mathbb{R})$ does not
hold. Indeed, if $Z\subset P$ is Lindelöf, choose $a\in\mathbb{R}$ such that
$\mathbb{R}_{a}\cap Z=\varnothing$ and set
$W=Z\cup\mathbb{R}_{a}\cup\mathbb{R}\times\mathbb{R}_{>0}$. Then $W$ is
Lindelöf and $f_{P}(P-Z)\ni a\not\in f_{P}(P-W)$.
We can obtain a boundaryless version by gluing copies of
$[0,1)\times\mathbb{R}$ to each boundary component $\mathbb{R}_{a}$ and
extending $f_{P}$ the obvious way. ∎

The following proposition can be proved by exactly the same argument that we
used in the proof of Lemma 5.4.

######  Proposition 5.7.

Let $X=\cup_{\alpha\in\omega_{1}}H_{\alpha}$ be a strict Lindelöf-chain (for
instance, a Type I space).
(a) If $H_{\alpha}$ is connected for uncountably many $\alpha$ then
$\mathsf{L}(X,\mathbb{R})$ holds. If $H_{\alpha}$ is moreover closed for
uncountably many $\alpha$, then $\mathsf{L_{cl}}(X,\mathbb{R})$ holds.
(b) If $X-H_{\alpha}$ is connected for uncountably many $\alpha$ then
$\mathsf{BR}(X,\mathbb{R})$ holds. If $H_{\alpha}$ is moreover closed for
uncountably many $\alpha$, then $\mathsf{BR_{cl}}(X,\mathbb{R})$ holds.

Finally, along the same lines we have:

######  Theorem 5.8.

Let $X$ be a space, $Y$ be a metrizable space and $\kappa$ be an infinite
cardinal.
(a) If $X$ is a strict compact-chain of length $\kappa$, then
$\mathsf{L_{cl}}(X,Y)$ holds.
(b) If $X$ is countably compact and a strict [open-and-contained-in-a-
Lindelöf-set]-chain of length $\kappa$, then $\mathsf{BR}(X,Y)$ holds.
(c) In particular, if $X$ is a countably compact Type I space, then both
$\mathsf{BR_{cl}}(X,Y)$ and $\mathsf{L_{cl}}(X,Y)$ hold.

###### Proof.

Let us write $X=\cup_{\alpha<\kappa}H_{\alpha}$ and fix $f:X\to Y$. There is
nothing to prove if $\kappa$ has countable cofinality because $X$ is then
Lindelöf. We thus assume that $\text{cf}(\kappa)$ is uncountable. Lemma 1.3
will be used again implicitely in what follows.
(a) Assume that each $H_{\alpha}$ is compact. Then $f(H_{\alpha})$ is also
compact; hence, Lemma 5.2 (a) implies that $f(H_{\alpha})=f(X)$ for some
$\alpha$.
(b) We may assume that $Y=f(X)$. By assumption $Y$ is compact and thus
separable (by Lemma 1.3). Since $H_{\alpha}$ is open, $X-H_{\alpha}$ is
countably compact, $f(X-H_{\alpha})$ is compact and thus closed. By Lemma 5.2
(b) the chain $U_{\alpha}=Y-f(X-H_{\alpha})$ cannot contain a strict cofinal
chain; hence, it must stagnate above some $\alpha$. It follows that
$f(X-H_{\alpha})=f(X-H_{\beta})$ for each $\beta\geq\alpha$. Notice that since
the $H_{\alpha}$ are open, any Lindelöf subset of $X$ must be contained in
some $H_{\gamma}$ (otherwise they form a cover without countable subcover).
Hence $\mathsf{BR}(X,Y)$ holds if $H_{\alpha}$ is contained in a Lindelöf set.
(c) If $X$ is of Type I,
$X=\cup_{\alpha<\omega_{1}}X_{\alpha}=\cup_{\alpha<\omega_{1}}\overline{X_{\alpha}}$.
By countable compactness $\overline{X_{\alpha}}$ is compact, hence
$\mathsf{L_{cl}}(X,Y)$ follows by (a). Lemma 2.2 and (b) imply that
$\mathsf{BR_{cl}}(X,Y)$ holds. ∎

We could have proved that $\mathsf{L_{cl}}(X,Y)$ holds in (c) with the
following lemma (since the image of $X$ in $Y$ is hereditarily separable).

######  Lemma 5.9.

Let $Y$ be hereditarily separable. If $X$ is an $\omega$-bounded space (in
particular, a Type I countably compact space), then $\mathsf{L_{cl}}(X,Y)$
holds and $f(X)$ is compact for any $f:X\to Y$.

###### Proof.

Given $f:X\to Y$, define $E\subset X$ by taking one preimage of each point in
a countable dense subset $D$ of $f(X)$. By $\omega$-boundedness $\overline{E}$
and thus $f(\overline{E})$ are compact, so in particular $f(\overline{E})$ is
closed. It follows that $f(\overline{E})=\overline{f(E)}=\overline{D}=f(X)$,
proving that $f(X)$ is compact and that $\mathsf{L_{cl}}(X,Y)$ holds. ∎

Notice that we cannot weaken the assumption to ‘$X$ is countably compact’, as
shown by the next example.

######  Example 5.10 (Folklore).

There is a countably compact space $X$ such that neither
$\mathsf{L_{cl}}(X,\mathbb{R})$ nor $\mathsf{BR_{cl}}(X,\mathbb{R})$ hold.

###### Details.

The idea is to obtain a countably compact non-Lindelöf space with a countable
dense subset of isolated points, and to apply Lemma 2.5. Let $\beta\omega$ be
the Čech-Stone compactification of the integers $\omega$. (The integers are
given the discrete topology.) The closure of any infinite set in $\beta\omega$
has cardinality $2^{\mathfrak{c}}$ (see e.g. [33, Lemma 0.1]), in particular
if $p\in\beta\omega-\omega$, then $\beta\omega-\\{p\\}$ is a countably compact
non-compact (since non-closed) subspace of $\beta\omega$ in which the integers
are dense. ∎

Under additional axioms, there are examples with more properties. For
instance, an Ostaszewski space (built with $\diamondsuit$, see [26]) is itself
a locally compact hereditarily separable countably compact non-compact space,
the identity map thus violates $\mathsf{L}(X,X)$. Example 2.8 (when
$\mathfrak{p}=\omega_{1}$) is another one with the same properties except
hereditary separability.

######  Question 5.11.

Is there is a countably compact space $X$ such that $\mathsf{L}(X,\mathbb{R})$
or $\mathsf{BR}(X,\mathbb{R})$ does not hold ?

Notice in passing that in Theorem 5.8 (c) and Lemma 5.9, the stronger property
$\mathsf{L_{cpt}}(X,Y)$ (defined in Section 9) holds.

##  6 A word on products on the target space

Let us have a quick look at which properties are easily seen to be preserved
under products in the target space. The first observation is obvious:

######  Lemma 6.1.

Let $\mathsf{P}$ be any property in Definition 1.2, $X$ and $Y_{j}$ be non
empty spaces for $j$ in some index set $J$. Then $\mathsf{P}(X,\prod_{j\in
J}Y_{j})\Rightarrow\mathsf{P}(X,Y_{k})$ for each $k\in J$.

###### Proof.

Each $Y_{k}$ is homeomorphic to a subspace of $\prod_{j\in J}Y_{j}$. Apply
Lemma 2.12. ∎

The next lemma is also almost immediate.

######  Lemma 6.2.

Let $X$ and $Y_{n}$, $n\in E$, be spaces.
(a) If $E$ is countable and $\mathsf{EC}(X,Y_{n})$ holds for each $n\in E$,
then $\mathsf{EC}(X,\prod_{n\in E}Y_{n})$ holds.
(b) If $E$ is finite and $\mathsf{EC_{cl}}(X,Y_{n})$ holds for each $n\in E$,
then $\mathsf{EC_{cl}}(X,\prod_{n\in E}Y_{n})$ holds.

###### Proof.

(a) Given $f:X\to\prod_{n\in E}Y_{n}$, for each $n\in E$ there is a Lindelöf
subspace $Z_{n}\subset X$ such that the projection $\pi_{n}\circ f:X\to Y_{n}$
is constant outside of $Z_{n}$. Hence $f$ is constant outside of the Lindelöf
set $\cup_{n\in E}Z_{n}$.
(b) As in (a). ∎

For $\mathsf{S}$, such a simple argument does not work even for $2$ factors,
as it is not clear a priori whether the retractions corresponding to each
projection can be combined. It is the case for $\omega_{1}$-trees, as seen in
Corollary 4.4 above.

######  Question 6.3.

Are there spaces $X,Y_{1},Y_{2}$ such that $\mathsf{S}(X,Y_{i})$ holds for
$i=1,2$ and $\mathsf{S}(X,Y_{1}\times Y_{2})$ does not ?

On the other hand, $\mathsf{L}$ and $\mathsf{BR}$ behave badly under products,
even when the domain space is a manifold.

######  Example 6.4.

Let $M$ be the surface

$\mathbb{L}_{\geq_{0}}\times[-2,1]\,-\,\cup_{\alpha\in\omega_{1}}\\{\alpha\\}\times[-1,1].$

Then $\mathsf{L_{cl}}(M,\mathbb{R})$ and $\mathsf{BR_{cl}}(M,\mathbb{R})$ hold
but $\mathsf{L}(M,\mathbb{R}^{2})$ and $\mathsf{BR}(M,\mathbb{R}^{2})$ do not.
Moreover, $\mathsf{S}(M,\mathbb{R})$ does not hold either.

Figure 2: Example 6.4.

###### Details.

Since $M$ is a manifold, $\mathsf{L}(M,\mathbb{R})$ holds by Theorem 5.1 above
and $\mathsf{L_{cl}}(M,\mathbb{R})$ follows since $M$ is Type I. Define
$M_{\alpha}$ to be the subset of points of $M$ with first coordinate
$<\alpha$. Since $M-M_{\alpha}$ is connected for each $\alpha$,
$\mathsf{BR}(M,\mathbb{R})$ (hence, $\mathsf{BR_{cl}}(M,\mathbb{R})$) holds by
Proposition 5.7 (b). To show that $\mathsf{L}(M,\mathbb{R}^{2})$ and
$\mathsf{BR}(M,\mathbb{R}^{2})$ do not hold, let
$\\{e_{\alpha}\,:\,\alpha\in\omega_{1}\\}$ (all distinct) be a subset of the
unit circle centered at the origin in $\mathbb{R}^{2}$. Let $I_{\alpha}$ be
the line segment joining the origin and $e_{\alpha}$. Define
$f:M\to\mathbb{R}^{2}$ as follows. If $x\in\mathbb{L}_{\geq 0}\times[-2,0]\cap
M$, $f(x)$ is the origin. If $x=\langle u,t\rangle$ with $\alpha<u<\alpha+1$
and $t\in(0,1]$, then $f(x)$ is the point of $I_{\alpha}$ at distance $t$ from
the origin. It is easy to check that $f$ is continuous and violates
$\mathsf{L}(M,\mathbb{R}^{2})$ and $\mathsf{BR}(M,\mathbb{R}^{2})$ since the
preimage of $I_{\alpha}$ minus the origin is $(\alpha,\alpha+1)\times(0,1]$.
To finish, observe that $\cup_{\alpha<\omega_{1}}(\alpha,\alpha+1)\times(0,1)$
is a discrete collection of open subsets of $M$ (see the first paragraph of
Section 8 for a reminder of the definition). We show in Section 8 (Lemma 8.5)
that this implies that $\mathsf{S}(M,\mathbb{R})$ (and the weaker property
$w\mathsf{S}(M,\mathbb{R})$ defined in Section 7) does not hold. ∎

##  7 Type I manifolds: when $\mathsf{EC}\not\Rightarrow\mathsf{S}$.

As written before, $\mathsf{EC}$ has the looks of a being stronger property
than $\mathsf{S}$, as being eventually constant is a rather strong assumption
for a map. But $\mathsf{S}$ asks for retractions, and some spaces do lack
retractions on smaller subspaces. Actually, even a weaker property (almost
stagnation, defined just next) might fail while $\mathsf{EC}$ holds. The next
definition could be subtitled “two new ways to procrastinate”.

######  Definition 7.1.

Given spaces $X,Y$, we say that $X$ almost stagnates (resp. weakly stagnates)
in $Y$, written $a\mathsf{S}(X,Y)$ (resp. $w\mathsf{S}(X,Y)$), iff for each
$f:X\to Y$ there is a Lindelöf $Z\subset X$ and $r:X\to Z$ such that $f\circ
r=f$ (resp. $f\circ r\upharpoonright(X-Z)=f\upharpoonright(X-Z)$).

We could have chosen $a\mathsf{S}$ as the “official” definition for stagnation
instead of the one with retractions, but all our counterexamples in this and
the next section are actually counter-examples to $a\mathsf{S}$ and not only
to $\mathsf{S}$, and Suslin trees in Section 4 do satisfy the stronger
property. The next lemma is immediate from the definitions (we again
abbreviate $\forall X,Y\,\mathsf{EC}(X,Y)$ simply by $\mathsf{EC}$, etc.)

######  Lemma 7.2.

${\mathsf{EC}}$${w\mathsf{S}}$${a\mathsf{S}}$${\mathsf{S}}$${\mathsf{L}}$

It is natural to ask whether some of these arrows do reverse. We already know
that $\mathsf{S}\not\Rightarrow\mathsf{EC}$; hence,
$w\mathsf{S}\not\Rightarrow\mathsf{EC}$ as well.

The main result of this section implies that
$\mathsf{EC}(M,\mathbb{R})\not\Rightarrow a\mathsf{S}(M,\mathbb{R})$ for $M$ a
manifold, and hence $w\mathsf{S}\not\Rightarrow a\mathsf{S}$ as well. As a
small warm up, let us show that some manifolds do lack retractions. This is
weaker than what we really need for the main theorem of this section, but the
proof contains one idea that will use again, which is contained in the
following lemma. It might be interesting to note that this is the only result
from basic algebraic topology that we need.

######  Lemma 7.3.

Let $\mathbb{S}^{1}=[0,1]/0\sim 1$ be the circle viewed as the interval with
identified endpoints. Let $C=[a,b]\times\mathbb{S}^{1}$ be the cylinder, let
$\pi:C\to\mathbb{S}^{1}$ be the projection on the second coordinate, and let
$d_{0},d_{1}:\mathbb{S}^{1}\to C$ be such that $\pi\circ d_{0}=\pi\circ
d_{1}=id_{\mathbb{S}^{1}}$. Let $I_{0}=[0,\frac{1}{3}]$,
$I_{1}=[\frac{1}{3},\frac{2}{3}]$ and $I_{1}=[\frac{2}{3},1]$, seen as subsets
of $\mathbb{S}^{1}$. Let $r:C\to C$ be given and set $s_{i}=\pi\circ r\circ
d_{i}:\mathbb{S}^{1}\to\mathbb{S}^{1}$, $i=0,1$. If $s_{0}(I_{k})\subset
I_{k}$ for $k=0,1,2$, then $s_{1}$ is not (homotopic to) a constant map.

###### Proof.

The assumptions imply that $s_{0}(I_{k})=I_{k}$. Sneak into the first few
classes of a course on the fundamental group to conclude. ∎

######  Proposition 7.4.

If $X=\cup_{\alpha\in\omega_{1}}X_{\alpha}$ is a longpipe such that
$\mathsf{EC}(X,\mathbb{R})$ holds, then $X$ does not retract on any
$\overline{X_{\alpha}}$.

###### Proof.

First, notice that $\mathsf{EC}(X,\mathbb{R}^{2})$ holds by Lemma 6.2. Suppose
that $r:X\to\overline{X_{\alpha}}$ is a retraction. Since
$\overline{X_{\alpha}}$ embeds in $\mathbb{R}^{2}$, for some $\gamma$ we have
$r(X-X_{\gamma})=\\{x\\}$. Since $r$ is the identity on
$\overline{X_{\alpha}}$, $\gamma>\alpha$. The boundary of
$\overline{X_{\alpha+1}}$ is homeomorphic to the circle (it might not be true
for $\overline{X_{\alpha}}$ though). Fix any successor $0<\beta\leq\alpha$.
The boundary of $\overline{X_{\beta}}$ is also homeomorphic to the circle.
Take a homeomorphism $\overline{X_{\gamma+1}}\to[0,3]\times\mathbb{S}^{1}$
that sends $\overline{X_{\beta}}$ and $\overline{X_{\gamma+1}}-X_{\gamma}$
respectively to $S_{1}=[0,1]\times\mathbb{S}^{1}$ and
$S_{2}=[2,3]\times\mathbb{S}^{1}$. This yields a retraction of the cylinder
$[0,3]\times\mathbb{S}^{1}$ into itself which is the identity on $S_{1}$ and
such that $S_{2}$ is sent to a point. But this is impossible by Lemma 7.3. ∎

The fact that the longpipes are ‘cylinders piled up’ is important in this
proposition. Each $X_{\alpha+1}$ in a longpipe is homeomorphic to
$[0,1)\times\mathbb{S}^{1}$, $X$ is thus a surface whose manifold boundary is
homeomorphic to $\mathbb{S}^{1}$. By sewing a disc in this ‘hole’, we obtain
what we call a sealed longpipe. Such a sealed longpipe $W$ has each
$\overline{W_{\alpha+1}}$ homeomorphic to the closed $2$-disc and its boundary
in any $W_{\beta}$ for higher $\beta$ is homeomorphic to the circle.

######  Lemma 7.5.

Let $X$ be a sealed longpipe, then $X$ retracts on $\overline{X_{\alpha+1}}$
$\forall\alpha\in\omega_{1}$.

In the remaining of this section, we let $B(a)$ denote the closed disk of
radius $a$ centered at the origin in $\mathbb{R}^{2}$.

###### Proof.

There is a homeomorphism $\overline{X_{\alpha+2}}\to B(2)$ that sends
$\overline{X_{\alpha+1}}$ to $B(1)$. Take a retraction of $B(2)$ on $B(1)$
which sends all the boundary of $B(2)$ to the origin. This yields a retraction
$r:\overline{X_{\alpha+2}}\to\overline{X_{\alpha+1}}$ such that the boundary
of $\overline{X_{\alpha+2}}$ is sent to a point $x\in X_{\alpha+1}$. Extend it
to the whole $X$ by $r(y)=x$ for all $y\not\in\overline{X_{\alpha+2}}$. ∎

However, the fact that there are retractions ‘as high as one wants’ does not
ensure that $\mathsf{EC}$ implies $a\mathsf{S}$. Actually, quite the opposite
is true as the main result of this section shows.

######  Theorem 7.6.

Let $X$ be either a longpipe or a sealed longpipe. If
$\mathsf{EC}(X,\mathbb{R})$ holds, then $a\mathsf{S}(X,\mathbb{R})$ does not
hold.

Before proving this theorem, let us show that there are concrete examples of
such (sealed) longpipes. We already encontered one: Theorems 3.3 and 7.6 show
that the longpipe built with $\clubsuit_{C}$ in Example 3.16 does not satisfy
$\mathsf{S}(Y,\mathbb{R})$. But there are examples in ZFC.

######  Example 7.7 (Nyikos, in effect).

There are sealed longpipes $X=\cup_{\alpha\in\omega_{1}}X_{\alpha}$ as in
Theorem 7.6 and Lemma 7.5. That is, $X$ retracts on each
$\overline{X_{\alpha+1}}$, $\mathsf{EC}(X,\mathbb{R})$ holds but
$a\mathsf{S}(X,\mathbb{R})$ does not.

###### Details.

This example is also due to Nyikos and is described in [24, p. 210]. It is
actually a longpipe, but we may sew a disc in $X_{1}$ to obtain a sealed one.
As in Example 3.5 to which we refer for definitions, we first consider a
tangent bundle $T\mathbb{L}_{+}$ of $\mathbb{L}_{+}$ given by some smoothing.
Write $\pi:T^{+}\to\mathbb{L}_{+}$ for the bundle projection. As written in
Example 3.5, in principle, the details of how the smoothing is built are
important, since $T\mathbb{L}_{+}$ and $T^{+}$ may have quite different
topological properties depending on the particular construction. This is
however irrelevant in our case, as we only use the fact that maps
$T^{+}\to\mathbb{R}$ are constant on the fibers above some some club
$C\subset\mathbb{L}_{+}$, which is true for any smoothing [24, Corollary
4.15]. Now consider the $\mathbb{Z}$-action on the fibers $\langle
x,i\rangle\mapsto 2^{i}\cdot x$ (see [24, p. 210]). Quotienting by this
action, we obtain a longpipe $X$ (each fiber is now a circle), and the
quotient map $q:T^{+}\to X$ is actually a covering. There is thus a unique
$\widetilde{\pi}:X\to\mathbb{L}_{+}$ such that the left part of the diagram
below commutes.

${\mathbb{L}_{+}}$${T^{+}}$${\mathbb{R}}$${X}$$\scriptstyle{\pi}$$\scriptstyle{q}$$\scriptstyle{f}$$\scriptstyle{\widetilde{f}}$$\scriptstyle{\widetilde{\pi}}$

Notice that there is no unbounded map $\mathbb{L}_{+}\to X$ (otherwise it
could be lifted to $T^{+}$, see e.g. [13, Theorem 5, Chapter 2, Section 4]).
Now, given $\widetilde{f}:X\to\mathbb{R}$, $f=\widetilde{f}\circ
q:T^{+}\to\mathbb{R}$ is constant on the fibers above a club
$C\subset\mathbb{L}_{+}$ and hence so is $\widetilde{f}$. It follows that $X$
satisfies $\mathsf{I0}$. By Theorem 3.8 (c) $\mathsf{EC}(X,\mathbb{R})$ holds.
Hence by Theorem 7.6 $a\mathsf{S}(X,\mathbb{R})$ does not hold. ∎

The proof of Theorem 7.6 is done by exhibiting a function $f:X\to\mathbb{R}$
such that any $r$ with $f\circ r=f$ has “wrong” homotopy properties; that is,
we may apply Lemma 7.3 (as in Proposition 7.4). For this purpose, we will
define families of maps such that if $f_{0},f_{1}$ are distinct member of one
family, then $f_{0}\circ r\not=f_{1}$ for any $r$ with relevant domain and
range. Our first proof used a kind of construction in three stages. At that
time we found convenient to try to be a bit systematic in our treatment, to
use (commutative) diagrams and a general abstract lemma (with an almost
trivial proof) enabling us to pass from a stage to the next. We then noticed
that a much simpler construction was available in just two stages. We could
have given the direct argument but chosed to keep first approach with the
abstract lemma since we find that it separates the proof intomore transparent
steps. So, let us state and prove our abstract lemma.

######  Lemma 7.8.

Let $Z_{0},Z_{1},X,Y$ be spaces and
$f_{0},f_{1},\widetilde{f_{0}},\widetilde{f_{1}},\psi_{0},\psi_{1},\varphi$ be
as in the diagram below, with $\widetilde{f_{0}}=f_{0}\circ\psi_{0}$,
$\widetilde{f_{1}}=f_{1}\circ\psi_{1}$ and $\psi_{1}\circ\varphi=id_{X}$.

${Z_{0}}$${X}$${Z_{1}}$${Y}$$\scriptstyle{\psi_{0}}$$\scriptstyle{\widetilde{f_{0}}}$$\scriptstyle{\varphi}$$\scriptstyle{f_{0}}$$\scriptstyle{f_{1}}$$\scriptstyle{\psi_{1}}$$\scriptstyle{\widetilde{f_{1}}}$

If there is no $r:X\to X$ such that $f_{0}\circ r=f_{1}$, then there is no
$\widetilde{r}:Z_{1}\to Z_{0}$ such that
$\widetilde{f_{0}}\circ\widetilde{r}=\widetilde{f_{1}}$.

###### Proof.

If $\widetilde{f_{0}}\circ\widetilde{r}=\widetilde{f_{1}}$ for some
$\widetilde{r}:Z_{1}\to Z_{0}$, set
$r=\psi_{0}\circ\widetilde{r}\circ\varphi$, then $r:X\to X$ and

$f_{0}\circ
r=f_{0}\circ\psi_{0}\circ\widetilde{r}\circ\varphi=\widetilde{f_{0}}\circ\widetilde{r}\circ\varphi=\widetilde{f_{1}}\circ\varphi=f_{1}\circ\psi_{1}\circ\varphi=f_{1},$

a contradiction. ∎

Figure 3: The family of interval maps $\\{f_{u}\,:\,u\in(0,\frac{1}{2})\\}$.

Our building block is a simple family of interval maps whose idea was given to
us by D. Gauld. In the remaining of this section, we denote the closed unit
interval $[0,1]$ by $I$. Let $u\in(0,\frac{1}{2})$. We define $f_{u}:I\to I$
to be the map (depicted in Figure 3, left) that takes values
$0,\frac{1}{2}+u,\frac{1}{2}-u,1$ at $0,\frac{1}{3},\frac{2}{3},1$,
respectively, and is linear in between.

######  Lemma 7.9 (D. Gauld).

Let $0<u,v<\frac{1}{2}$ with $u\not=v$. Then there is no continuous $r:I\to I$
such that $f_{u}\circ r=f_{v}$.

###### Proof.

Suppose $u>v$. Then $r$ must be increasing up to $1/3$ but $r(1/3)<1/3$, then
since $f_{v}$ must decrease, $r$ is decreasing until $2/3$, then again
increasing with $r(1)=1$. But we run into problems when $r(x)$ reaches $1/3$
since then $f_{u}\circ r$ starts decreasing while $f_{v}$ is not. Suppose now
that $u<v$. Then we run into problems as soon as $r(x)$ reaches $1/3$. ∎

Let $i:I\to B(1)$, $j:B(1)\to I$ be defined as $i(t)=\langle 1-t,0\rangle$ and
$j(x)=1-|x|$ (where $|x|$ is the Euclidean norm). Define
$\widetilde{f_{u}}:B(1)\to I$ as $\widetilde{f_{u}}=f_{u}\circ j$.

######  Corollary 7.10.

$\exists\widetilde{r}:B(1)\to B(1)$ with
$\widetilde{f_{u}}\circ\widetilde{r}=\widetilde{f_{v}}$ iff $u=v$.

###### Proof.

The result follows immediately by the fact that $j\circ i=id_{I}$, the diagram
below and Lemmas 7.8 and 7.9.

${B(1)}$${{I}}$${B(1)}$${{I}}$$\scriptstyle{j}$$\scriptstyle{\widetilde{f_{u}}}$$\scriptstyle{i}$$\scriptstyle{p_{n}}$$\scriptstyle{p_{m}}$$\scriptstyle{j}$$\scriptstyle{\widetilde{f_{v}}}$

∎

The family of maps $\widetilde{f_{u}}$ enables us to prove the following.

######  Lemma 7.11.

Let $C=[0,2]\times\mathbb{S}^{1}$ be the cylinder, and set
$B=B(2)\subset\mathbb{R}^{2}$. Then the following hold.
(a) There is $f:C\to I$, constant on $[1,2]\times\mathbb{S}^{1}$ such that if
$r:C\to C$ satisfies $f\circ r=f$, then
$r\upharpoonright\\{2\\}\times\mathbb{S}^{1}$ is not (homotopic to) a constant
map.
(b) There is $g:B\to I$, constant on $\\{x\in B\,:\,|x|\geq 1\\}$, such that
if $s:B\to B$ satisfies $g\circ s=g$, then $s$ is not constant on $\partial
B=\\{x\in B\,:\,|x|=2\\}$.

Figure 4: The subsets for defining the functions $f:C\to I$ and $g:B\to I$.

###### Proof.

(a) We see $\mathbb{S}^{1}$ again as the interval $I$ with endpoints
identified. Let $E$, $D_{0}$, $D_{1}$ and $D_{2}$ be the closed regions
(boundaries are included) of $C$ depicted in Figure 4 (left). Fix any
homeomorphisms $\phi_{k}:D_{k}\to B(1)$, $k=0,1,2$. Fix three distincts points
$u_{0},u_{1},u_{2}\in(0,\frac{1}{2})$. Define $f:C\to I$ to be constant on $0$
on $E$ and equal to $\widetilde{f}_{u_{k}}\circ\phi_{k}$ on $D_{k}$, for
$k=0,1,2$. By definition, $\widetilde{f}_{u_{k}}\circ\phi_{k}$ takes value $0$
on the boundary of $D_{k}$. It follows that $f$ is continuous. Let $r:C\to C$
be such that $f\circ r=f$. Since $f^{-1}(\\{0\\})$ is equal to $E$ union the
boundaries of $D_{0}$ and $D_{1}$, by connectedness $r$ must send the interior
of $D_{0}$ to the interior of $D_{0}$, $D_{1}$ or $D_{2}$. Hence,
$r\upharpoonright D_{0}$ has range included in $D_{k}$ where $k$ is $0$, $1$
or $2$. But we are in the following situation:

${D_{0}}$${B(1)}$${D_{k}}$${{I}}$$\scriptstyle{\phi_{0}}$$\scriptstyle{\widetilde{f}_{u_{0}}\circ\phi_{0}}$$\scriptstyle{\phi_{k}^{-1}}$$\scriptstyle{\widetilde{f}_{u_{0}}}$$\scriptstyle{\widetilde{f}_{u_{k}}}$$\scriptstyle{\phi_{k}}$$\scriptstyle{\widetilde{f}_{u_{k}}\circ\phi_{k}}$

By Lemma 7.8 and Corollary 7.10, we must have $k=0$, and $r$ sends thus
$D_{0}$ into itself. The same argument shows that $r$ sends also $D_{1}$ and
$D_{2}$ respectively into themselves. Let $d:\mathbb{S}^{1}\to C$ be
$d(x)=\langle\frac{1}{2},x\rangle$ and $\pi:C\to\mathbb{S}^{1}$ be the
projection on the second coordinate. By definition $\pi\circ r\circ d$ sends
the intervals
$[0,\frac{1}{3}],[\frac{1}{3},\frac{2}{3}],[\frac{2}{3},1]\subset\mathbb{S}^{1}$
respectively into themselves. By Lemma 7.3, $r$ cannot be (homotopic to) a
constant map on $\\{2\\}\times\mathbb{S}^{1}$.
(b) The proof is almost the same, but this time we define $E$, $D_{k}$
($k=0,1,2,3$) as in the righthandside of Figure 4. Similarly as in (a), take
four distinct points $u_{0},u_{1},u_{2},u_{3}\in(0,\frac{1}{2})$, fix
homeomorphisms $\phi_{k}:D_{k}\to B(1)$, and define $g$ exactly as $f$. Then,
argue as in (a) to show that if $g\circ s=g$, then $s$ sends $D_{1}\cup
D_{2}\cup D_{3}$ to itself. By restricting to this subset, we are exactly in
the same situation than in (a), and we may conclude. ∎

We may now prove Theorem 7.6.

###### Proof of Theorem 7.6.

Let $C$ be the cylinder $[0,2]\times\mathbb{S}^{1}$ and $f:C\to\mathbb{R}$ be
given by Lemma 7.11 (a). If $X=\cup_{\alpha<\omega_{1}}X_{\alpha}$ is a
longpipe, fix a homeomorphism $\Psi:\overline{X_{1}}\to C$ and let
$\widehat{f}$ be defined as $f\circ\Psi$ on $\overline{X_{1}}$ and constant on
$0$ elswhere. If there is $r:X\to\overline{X_{\alpha}}$ such that
$\widehat{f}\circ r=\widehat{f}$, then since $\overline{X_{\alpha}}$ embeds in
$\mathbb{R}^{2}$, by $\mathsf{EC}(X,\mathbb{R})$, there is some $\beta$ such
that $r$ is constant on $X-X_{\beta}$. We may assume that $\beta>\alpha$. Then
$r\upharpoonright\overline{X_{\beta+1}}$ is a map sending a cylinder into
itself and constant on the upper boundary. But this is impossible by Lemma
7.11 (a). If $X$ is a sealed longpipe, the proof is the same, using $g$ given
by Lemma 7.11 (b). ∎

We end this section with a last example. The proofs of the claimed properties
are left to the reader, as they are very similar to what we just did.

######  Example 7.12.

Let $P^{0},P^{1}$ be longpipes such that $\mathsf{EC}(P^{k},\mathbb{R})$ hold
for $k=0,1$. Let $P$ be obtained by gluing $P^{0}$ and $P^{1}$ along their
boundary (which is homeomorphic to a circle) in any way. Then
$w\mathsf{S}(P,\mathbb{R})$ holds but $a\mathsf{S}(P,\mathbb{R})$ and
$\mathsf{EC}(P,\mathbb{R})$ do not.

##  8 $w\mathsf{S}(M,\mathbb{R})$, normality, collectionwise Hausdorfness and
$\omega_{1}$-compactness for manifolds

Recall that a collection of subsets of a space $X$ is discrete iff for each
$x\in X$ there is an open set containing it which intersects at most one
member of the collection. In a discrete collection
$\\{U_{\alpha}\,:\,\alpha\in\lambda\\}$, $\cup_{\alpha\in
K}\overline{U_{\alpha}}$ is closed for any $K\subset\lambda$. Aspace is
$\kappa$-[strongly] collectionwise Hausdorff (abbreviated $\kappa$-[s]cwH) iff
any closed discrete subset $\\{x_{\alpha}\,:\,\alpha\in\lambda\\}\subset X$ of
size $\lambda\leq\kappa$ can be expanded to a disjoint [resp. discrete]
collection of open sets $\\{U_{\alpha}\,:\,\alpha\in\lambda\\}$ with
$x_{\alpha}\in U_{\alpha}$. If $X$ is $\kappa$-[s]cwH for each $\kappa$, we
say that $X$ is [s]cwH.

It happens that, in the world of manifolds, $\mathsf{S}(X,\mathbb{R})$ and its
weakenings $a\mathsf{S}(X,\mathbb{R})$ and $w\mathsf{S}(X,\mathbb{R})$ (see
Definition 7.1) have an interesting interplay with normality, collectionwise
Hausdorfness and $\omega_{1}$-compactness. Let us first give an example.

######  Example 8.1.

(Similar to [4, Ex. 5.5])
Let $X=\mathbb{L}_{\geq 0}\times\mathbb{R}-\omega_{1}\times\\{0\\}$. Then $X$
is a cwH non-normal and non-$\omega_{1}$-compact surface, and
$\mathsf{S}(X,\mathbb{R})$ holds.

Here, of course, $\omega_{1}$ is seen as a subspace of $\mathbb{L}_{\geq 0}$.
All the properties are proved as in [4, Ex. 5.5] except
$\mathsf{S}(X,\mathbb{R})$ which is proved exactly as in Example 2.10.

######  Question 8.2.

If $M$ is a normal manifold such that $\mathsf{S}(M,\mathbb{R})$ (or
$a\mathsf{S}(M,\mathbb{R})$, or $w\mathsf{S}(M,\mathbb{R})$) hold, is then $M$
$\omega_{1}$-compact ?

Notice that this question asks for a partial generalization of Theorem 3.4
(where we have $\mathsf{EC}$ instead of $\mathsf{S}$ or its weakenings). The
next theorem is a partial answer for the $a\mathsf{S}$-version.

######  Theorem 8.3.

Let $M$ be an $\aleph_{1}$-scwH manifold. If $w\mathsf{S}(M,\mathbb{R})$ holds
then $M$ is $\omega_{1}$-compact.

Recall that a normal $\kappa$-cwH space is $\kappa$-scwH, so Theorem 8.3
answers Problem 8.2 if the following old open problem – which is probably more
fundamental than anything done in the present paper and already appeared
various times in print – has an affirmative answer:

######  Problem 8.4.

Is every normal manifold $\aleph_{1}$-scwH ?

As far as we know, there is no consistent counter-example to Problem 8.4, and
the answer is affirmative under V=L or in any model obtained after forcing
with a Suslin tree, see [32].

For the proof of Theorem 8.3 we will use the fonctions
$\widetilde{f_{u}}=f_{u}\circ j$, where $f_{u}$ is defined in Figure 3 and $j$
just before Corollary 7.10. The bulk of the argument for proving Theorem 8.3
is done in the next lemma.

######  Lemma 8.5.

Let $M$ be a manifold and $D$ be a discrete collection of open sets in $M$. If
$D$ is uncountable, then $w\mathsf{S}(M,\mathbb{R})$ does not hold.

###### Proof.

Let $B$ be the closed unit ball in $\mathbb{R}^{n}$, where $n$ is the
dimension of $M$. We may assume that
$D=\\{N_{\alpha}\,:\,\alpha\in\omega_{1}\\}$ and that each $N_{\alpha}$ is
homeomorphic to $B$. Fix a homeomorphism $\psi_{\alpha}:B\to N_{\alpha}$. Fix
a (non-continuous) $1$-to-$1$ map $\varphi:\omega_{1}\to(0,\frac{1}{2})$.
Define $h:M\to[0,1]$ as follows. On the complement of the union of the
interiors of the $N_{\alpha}$, $h$ takes value $0$. Then, define
$h\upharpoonright N_{\alpha}$ as
$\widetilde{f}_{\varphi(\alpha)}\circ\psi_{\alpha}^{-1}$. By construction $h$
is $0$ on the boundary of $N_{\alpha}$; hence, $h$ is continuous by
discreteness of the $N_{\alpha}$’s.
Suppose that there is some Lindelöf $Z\subset M$ and $r:M\to Z$ such that
$h\circ r\upharpoonright(X-Z)=h\upharpoonright(X-Z)$. Then $Z$ intersects at
most countably many $N_{\alpha}$ by Lindelöfness. Fix $\alpha$ such that
$Z\cap N_{\alpha}=\varnothing$. Then, by connectedness, the image under $r$ of
$N_{\alpha}$ must be contained in some $N_{\beta}$, and we have the following
diagram.

${N_{\alpha}}$${B}$${N_{\beta}}$${{I}}$$\scriptstyle{\psi_{\alpha}^{-1}}$$\scriptstyle{\widetilde{f}_{\varphi(\alpha)}}$$\scriptstyle{\psi_{\beta}^{-1}}$$\scriptstyle{f_{\varphi(\alpha)}}$$\scriptstyle{f_{\varphi(\beta)}}$$\scriptstyle{\psi_{\beta}}$$\scriptstyle{\widetilde{f}_{\varphi(\beta)}}$

This yields a contradiction by Lemmas 7.8 and 7.9. ∎

###### Proof of Theorem 8.3.

Suppose there is an uncountable closed discrete subset $D$, up to taking a
subset, we may assume that $|D|=\aleph_{1}$. Expand $D$ to a discrete
collection of closed neighborhoods $\\{N_{w}\,:\,w\in D\\}$. Then apply Lemma
8.5. ∎

Note in passing that since any countable subset of a manifold is contained in
an open set homeomorphic to $\mathbb{R}^{n}$ (see [11, Cor. 3.4]), an
$\omega_{1}$-compact manifold is cwH. But it may fail to be scwH (at least
consistently), see Example 9.6 below, which is moreover non-normal. This space
is eventually constant in $\mathbb{R}$. Another non-normal manifold $M$ such
that $\mathsf{EC}(M,\mathbb{R})$ holds, but Type I this time, is Example 3.5.

##  9 If ‘small’ means ‘compact’

In this brief section, we look at the properties obtained if every instance of
‘Lindelöf’ is replaced by ‘compact’ in Definition 1.2. We call the
corresponding properties $\mathsf{P_{cpt}}$ for
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{S},\mathsf{L},\mathsf{BR}\\}$. Let us
first gather some trivial facts in a lemma.

######  Lemma 9.1.

Let $\mathsf{P}\in\\{\mathsf{EC},\mathsf{S},\mathsf{L},\mathsf{BR}\\}$. Then
the following hold.
(a)
$\mathsf{BR_{cpt}}\Longleftarrow\mathsf{EC_{cpt}}\Longrightarrow\mathsf{L_{cpt}}\Longleftarrow\mathsf{S_{cpt}}$.
(b) $\mathsf{P_{cpt}}\Longrightarrow\mathsf{P_{cl}}$.
(c) If $X$ is countably compact and $Y$ any space then
$\mathsf{P_{cl}}(X,Y)\Longleftrightarrow\mathsf{P_{cpt}}(X,Y)$.
(d) If $\mathsf{L_{cpt}}(X,\mathbb{R})$ holds then $X$ is pseudocompact.

By (d) and Theorem 5.1, any Type I non pseudocompact manifold satisfies
$\mathsf{L_{cl}}(X,\mathbb{R})$ but not $\mathsf{L_{cpt}}(X,\mathbb{R})$. For
instance, $\mathsf{S_{cl}}(\mathbb{L}_{+},\mathbb{R})$ and
$\mathsf{BR_{cl}}(\mathbb{L}_{+},\mathbb{R})$ hold, but their cpt-counterparts
do not. We note also that the converse of point (d) does not hold for
Hausdorff spaces:

######  Example 9.2 ([30], Ex. 3.1).

Let $[0,1]_{s}$ be the space obtained by refining the Euclidean topology on
$[0,1]$ letting $\\{1/n\,:\,n\in\omega\\}$ be closed. Then $[0,1]_{s}$ is
pseudocompact and non-compact, but $\mathsf{L_{cpt}}([0,1]_{s},\mathbb{R})$
does not hold, as shown by $id:[0,1]_{s}\to[0,1]$.

In view of points (b)–(d) above, it seems interesting to see whether
$\mathsf{P_{cl}}(X,Y)\Rightarrow\mathsf{P_{cpt}}(X,Y)$ when $X$ is
pseudocompact. Recall that a normal pseudocompact space is countably compact
and a Lindelöf regular space is normal [10, Thms 3.10.21 & 3.8.2]. A closed
subset of a pseudocompact space may fail to be pseudocompact, but the
following is well known (see e.g. [30, p. 447]:

######  Lemma 9.3.

Let $U$ be an open subset of a Tychonoff pseudocompact space $X$. Then
$\overline{U}$ is pseudocompact.

This lemma gives almost immediately the following theorems. We first show that
the situation is quite simple for Type I spaces.

######  Theorem 9.4.

Let $X=\cup_{\alpha\in\omega_{1}}X_{\alpha}$ be a regular Type I pseudocompact
space and $Y$ be any space. Then $X$ is countably compact, and thus
$\mathsf{P}(X,Y)\Longleftrightarrow\mathsf{P_{cpt}}(X,Y)$ for each
$\mathsf{P}\in\\{\mathsf{EC},\mathsf{S},\mathsf{L},\mathsf{BR}\\}$.

###### Proof.

In a regular Type I space $X$, each $\overline{X_{\alpha}}$ is Lindelöf
regular and hence normal. It follows that $X$ is Tychonoff. Hence, each
$\overline{X_{\alpha}}$ is pseudocompact by Lemma 9.3 and thus compact. If $X$
is not countably compact, then there is a countable closed discrete subset $D$
in $X$. Then, for some $\alpha$, $D$ is contained in $\overline{X_{\alpha}}$
which is a contradiction. Conclude with Lemmas 2.2 and 9.1 (c). ∎

When $X$ is not of Type I, we can still say something for $\mathsf{EC}$.

######  Theorem 9.5.

Let $X$ be a Tychonoff pseudocompact space and $Y$ be any space. Then

$\mathsf{EC_{cl}}(X,Y)\Longleftrightarrow\mathsf{EC_{cpt}}(X,Y).$

###### Proof.

The reverse implication is immediate. Suppose that $\mathsf{EC_{cl}}(X,Y)$
holds. Given $f:X\to Y$, there is some $y\in Y$ such that
$U=f^{-1}(Y-\\{y\\})$ is contained in a closed Lindelöf subset $Z$ of $X$.
Then $Z$ is (regular Lindelöf hence) normal, and $\overline{U}\subset Z$ as
well. By Lemma 9.3, $\overline{U}$ is countably compact and hence compact. By
definition, $f$ is constant outside of $\overline{U}$. This shows that
$\mathsf{EC_{cpt}}(X,Y)$ holds. ∎

We note that the implication
$\mathsf{EC}(X,\mathbb{R})\Rightarrow\mathsf{EC_{cpt}}(X,\mathbb{R})$ for
pseudocompact $X$ does not hold, as shown by Example 2.7. If
$\mathfrak{b}=\omega_{1}$, there is even a manifold counter-example. (This
example was also alluded to in Section 2.) Recall that $\mathfrak{b}$ is the
smallest cardinality of an $<^{*}$-unbounded family of functions
$\omega\to\omega$, where $f<^{*}g$ iff there is some $n\in\omega$ such that
$f(m)<g(m)$ when $m\geq n$. We can assume that such an unbounded family is
well ordered by $<^{*}$ (see e.g. [8, Theorem 3.3]). Recall in passing that
$\omega_{1}\leq\mathfrak{p}\leq\mathfrak{b}\leq 2^{\omega}$ and that each
inequality may be strict depending on the model of set theory.

######  Example 9.6 (Nyikos [22, Ex. 6.3]).

An $\omega_{1}$-compact surface $S$ which is not countably compact, such that
$\mathsf{EC}(X,\mathbb{R})$ holds but $\mathsf{EC_{cpt}}(X,\mathbb{R})$ and
$\mathsf{EC_{cl}}(X,\mathbb{R})$ do not. Moreover, $\mathsf{BR}(X,\mathbb{R})$
and $\mathsf{L}(X,\mathbb{R})$ hold. If $\mathfrak{b}=\omega_{1}$, then $S$
can be made pseudocompact and not scwH.

###### Idea of the construction.

We only give a sketch, as the general construction is detailed in [11, Example
1.29], and appeared for the first time in [22, Ex. 6.3]. The idea is quite
similar to (a version) of Example 2.8. Start with an $<^{*}$-unbounded family
of functions $f_{\alpha}:\omega\to\omega$, $\alpha\in\mathfrak{b}$. We might
assume that each $f_{\alpha}$ is strictly increasing and $f_{\alpha}(0)=0$.
Now consider (the graphs of) the strictly increasing maps $[0,1)\to[0,1)$
(with supremum $1$) given by first embedding $\omega\times\omega$ in
$[0,1)^{2}$, sending $\langle n,m\rangle$ to $\langle
1-\frac{1}{n+1},1-\frac{1}{m+1}\rangle$ and interpolating linearly in between
to obtain maps $\widehat{f}_{\alpha}:[0,1)\to[0,1)$. The surface can then be
seen as the unit square $[0,1]^{2}$ with $\langle 1,1\rangle$ removed, to
which is attached a copy of $\mathbb{L}_{\geq 0}$ in such a way that
$\lim_{x\to 1}\widehat{f}_{\alpha}(x)=\alpha\in\mathbb{L}_{\geq 0}$ when
$\alpha<\omega_{1}$. (The actual construction by Nyikos is actually slightly
different, but only on a superficial level.) $\mathsf{EC}(S,\mathbb{R})$ holds
because any real valued map on $\mathbb{L}_{\geq 0}$ is eventually constant
and the remainder of the space is Lindelöf. The construction is made such that
the subset $[0,1)\times\\{1\\}$ is closed (hence $S$ is not countably
compact), and $\\{1\\}\times[0,1)$ is “attached” at the start of the copy of
$\mathbb{L}_{\geq 0}$. Since any uncountable subset has a cluster point either
in $[0,1]^{2}-\\{\langle 1,1\rangle\\}$ or in $\mathbb{L}_{\geq 0}$, $S$ is
$\omega_{1}$-compact.
The real valued map consisting of the projection on the second factor on
$[0,1]^{2}-\\{\langle 1,1\rangle\\}$ and constant on $1$ on $\mathbb{L}_{\geq
0}$ contradicts $\mathsf{EC_{cl}}(X,\mathbb{R})$. Since $S$ is a strict chain
given by $([0,1]^{2}-\\{\langle 1,1\rangle\\})\cup[0,\alpha)$ (where the
interval is a subset of $\mathbb{L}_{\geq 0}$), $\mathsf{BR}(S,\mathbb{R})$
holds by Proposition 5.7, and $\mathsf{L}(S,\mathbb{R})$ holds by Theorem 5.1.
If $\mathfrak{b}>\omega_{1}$, then $\lim_{x\to 1}f_{\alpha}(x)$ does not exist
when $\alpha\geq\omega_{1}$, but if on the contrary $\mathfrak{b}=\omega_{1}$,
Nyikos showed that the resulting surface is pseudocompact. Indeed, in this
case, any countable sequence in $[0,1]\times[0,1)$ has a cluster point.
Moreover, the closed discrete subspace $\\{\langle
1-1/n,1\rangle\,:\,n\in\omega\\}$ cannot be expanded to a discrete open
collection, because taking one point in each open set intersected with
$[0,1]\times[0,1)$ yields a cluster point. Hence, $S$ is not scwH. ∎

Notice that $w\mathsf{S}(S,\mathbb{R})$ holds since
$\mathsf{EC}(S,\mathbb{R})$ holds. Whether $a\mathsf{S}(S,\mathbb{R})$,
$\mathsf{S}(S,\mathbb{R})$, $\mathsf{L_{cl}}(S,\mathbb{R})$ or
$\mathsf{BR_{cl}}(S,\mathbb{R})$ hold is less clear.

##  10 Summary and tables

In this section, we summarize some of our results in a concise (albeit
incomplete) form. First, and just for the pleasure of drawing an overly
complicated diagram, the following implications hold by Lemmas 2.1, 7.2 and
9.1 (whose proofs are all immediate). Recall that by
$\mathsf{P}_{1}\Rightarrow\mathsf{P}_{2}$ we mean $\forall
X,Y\,\mathsf{P}_{1}(X,Y)\Rightarrow\mathsf{P}_{2}(X,Y)$.

${\mathsf{S_{cpct}}}$${a\mathsf{S_{cpct}}}$${w\mathsf{S_{cpct}}}$${\mathsf{EC_{cpct}}}$${\mathsf{BR_{cpct}}}$${\mathsf{L_{cpct}}}$${\mathsf{S_{cl}}}$${a\mathsf{S_{cl}}}$${w\mathsf{S_{cl}}}$${\mathsf{EC_{cl}}}$${\mathsf{BR_{cl}}}$${\mathsf{L_{cl}}}$${\mathsf{S}}$${a\mathsf{S}}$${w\mathsf{S}}$${\mathsf{EC}}$${\mathsf{BR}}$${\mathsf{L}}$

If the domain space is Type I, the vertical arrows between $\mathsf{P}$ and
$\mathsf{P_{cl}}$ reverse (Lemma 2.2) and if the domain space is Type I and
countably compact, the vertical arrows between $\mathsf{P_{cl}}$ and
$\mathsf{P_{cpt}}$reverse (Theorem 9.4). The thick red arrows are the only
ones for which we do not know a counterexample to their converse, as shown by
the tables below. These red arrows yield the following questions that an
attentive reader had probably already formulated in their head, and maybe even
solved, which is not our case.

######  Question 10.1.

Are there spaces $X,Y$ such that $a\mathsf{S}(X,Y)$ holds but not
$\mathsf{S}(X,Y)$ ? Is there an example with $Y=\mathbb{R}$ and/or $X$ a
manifold ?

######  Question 10.2.

Are there spaces $X,Y$ such that $a\mathsf{S}(X,Y)$ holds but not
$a\mathsf{S_{cl}}(X,Y)$ ? Is there an example with $Y=\mathbb{R}$ and/or $X$ a
manifold ?

The tables below compile most of our (counter-)examples (and some positive
results). We believe that our choice of notation is self explanatory. Table 1
summarizes quickly some of the results of section 3. In the same vein, the
contents of section 4 and some of section 5 are summarized in the Tables 2–3.
Finally, Tables 4–5 scan through the whole paper for (counter)-examples to
$\mathsf{P}(X,\mathbb{R})$. There, we let $K_{0}$ and $K_{1}$ be the spaces
described in Examples 7.7 and 7.12. We denote the discrete space of
cardinality $\aleph_{1}$ by $D_{\aleph_{1}}$. The question mark “?” means that
we do not know whether the property in question holds (it might however not be
difficult to find out). The fact that $a\mathsf{S}(X,\mathbb{R})$ does not
hold for Examples 2.7–2.8 (Table 4) follows easily with the same maps we used
to show that $\mathsf{L_{cl}}(X,\mathbb{R})$ does not hold. The fact that
$w\mathsf{S}(X,\mathbb{R})$ does not hold for Example 5.6 follows from Lemma
8.5 (Table 4). In Table 5, Lemma 9.1 (d) is responsible for the cpt-properties
not holding, since the spaces in question are not pseudocompact.

Reference | Axiom | Property of $Y$
---|---|---
isocompact | $G_{\delta}$ points | countably tight | loc. compact | $\mathsf{EC}(\omega_{1},Y)$
or Type I
Ex. 3.13 |  |  | $\times$ |  |  | $\times$
Ex. 3.16 | $\clubsuit_{C}$ | $\times$ |  |  |  |
Thm 3.15 | PFA | implied |  | assumed | assumed | assumed
Table 1: Section 3 – about $\mathsf{EC}(\omega_{1},Y)$ Reference | Properties of $Y$ |  $T$ $\omega_{1}$-cpct | $\mathsf{S}(T,Y)$ | $\mathsf{BR}(T,Y)$
---|---|---|---|---
Thm 4.2 & Prop 4.5 | submetrizable | assumed | implied | implied
Prop. 4.5 | uncountable |  | equivalent | equivalent
Ex. 4.6 | countable | $\times$ | $\times$ |
Ex. 4.7 | her. separable |  | $\times$ | $\times$
Table 2: Section 4 – $\mathsf{S}(T,Y)$ and $\mathsf{BR}(T,Y)$ when $T$ is an $\omega_{1}$-tree Reference |  $X$ is Type I | $\mathsf{S_{cl}}(X,\mathbb{R})$ | $\mathsf{BR_{cl}}(X,\mathbb{R})$
---|---|---|---
Thm 5.8 (c) | assumed | implied | implied
Ex. 5.10 | $\times$ | $\times$ | $\times$
Table 3: Section 5 – $\mathsf{BR_{cl}}(X,\mathbb{R})$ and $\mathsf{L_{cl}}(X,\mathbb{R})$ when $X$ is countably compact $X$ | Property $\mathsf{P}$ |  $X$ is a
---|---|---
$\mathsf{L}$ | $\mathsf{L_{cl}}$ | $\mathsf{BR}$ | $\mathsf{BR_{cl}}$ | $\mathsf{EC}$ | $\mathsf{EC_{cl}}$ | $w\mathsf{S}$ | $a\mathsf{S}$ | $\mathsf{S}$ | manifold
$D_{\aleph_{1}}$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\mathbb{L}_{+}$ |  |  |  |  |  |  |  |  |  |
$\mathbb{L}$ |  |  |  |  | $\times$ | $\times$ |  |  |  |
2.7 |  | $\times$ |  | $\times$ |  | $\times$ |  | $\times$ | $\times$ | $\times$
2.8 |  | $\times$ |  | $\times$ |  | $\times$ |  | $\times$ | $\times$ | $\times$
2.10 (a) |  |  | $\times$ | $\times$ | $\times$ | $\times$ |  |  |  | $\times$
2.10 (b) | $\times$ | $\times$ |  |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
3.5 |  |  |  |  |  |  |  |  |  |
5.6 |  | ? | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |
6.4 |  |  |  |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |
$K_{0}$ |  |  |  |  |  |  |  | $\times$ | $\times$ |
$K_{1}$ |  |  |  |  | $\times$ | $\times$ |  | $\times$ | $\times$ |
9.6 |  | ? |  | ? |  | $\times$ |  | ? | ? |
Table 4: $\mathsf{P}(X,\mathbb{R})$ and $\mathsf{P_{cl}}(X,\mathbb{R})$ $X$ | Property $\mathsf{P}$ |  $X$ is a
---|---|---
$\mathsf{L_{cl}}$ | $\mathsf{L_{cpt}}$ | $\mathsf{BR_{cl}}$ | $\mathsf{BR_{cpt}}$ | $\mathsf{EC_{cl}}$ | $\mathsf{EC_{cpt}}$ | $w\mathsf{S_{cl}}$ | $w\mathsf{S_{cpt}}$ | $a\mathsf{S_{cl}}$ | $a\mathsf{S_{cpt}}$ | $\mathsf{S_{cl}}$ | $\mathsf{S_{cpt}}$ | manifold
$\mathbb{L}_{\geq 0}$ |  |  |  |  |  |  |  |  |  |  |  |  | (with bdry)
$\mathbb{L}_{+}$ |  | $\times$ |  | $\times$ |  | $\times$ |  | $\times$ |  | $\times$ |  | $\times$ |
$\mathbb{L}_{+}\times\mathbb{R}$ |  | $\times$ |  | $\times$ | $\times$ | $\times$ |  | $\times$ |  | $\times$ |  | $\times$ |
$K_{0}$ |  |  |  |  |  |  |  |  | $\times$ | $\times$ | $\times$ | $\times$ |
$K_{0}-\\{pt\\}$ |  | $\times$ |  | $\times$ |  | $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |
$K_{1}$ |  |  |  |  | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |
$K_{1}-\\{pt\\}$ |  | $\times$ |  | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ |
Table 5: $\mathsf{P_{cl}}(X,\mathbb{R})$ and $\mathsf{P_{cpt}}(X,\mathbb{R})$

## References

  * [1] P. Alexandroff and P. Urysohn.  Mémoire sur les espaces topologiques compacts.  Number 14 in Proceedings of the section of mathematical sciences. Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, 1929.
  * [2] P. Bacon.  The compactness of countably compact spaces.  Pac. J. Math., 32(3):587–592, 1970.
  * [3] M. Baillif.  Directions in Type I spaces.  preprint arXiv:1404.1398, 2014.
  * [4] M. Baillif.  Relative (functionally) Type I spaces and narrow subspaces.  To appear in Top. Proc., 2022.
  * [5] M. Baillif.  Non-metrizable manifolds and contractibility.  Preprint arXiv:2308.03459, 2023.
  * [6] Z. Balogh.  On compact Hausdorff spaces of countable tightness.  Proc. Amer. Math. Soc., 105:755–764, 1989.
  * [7] M.H. Cho and W.W. Park.  Isocompactness and related topics of weak covering property.  Bull. Korean Math. Soc., 39(2):347–357, 2002.
  * [8] E. K. Van Douwen.  The integers and topology.  In Handbook of Set-Theoretic Topology (Kenneth Kunen and Jerry E. Vaughan, eds.), pages 111–167. North-Holland, Amsterdam, 1984.
  * [9] T. Eisworth.  On perfect pre-images of $\omega_{1}$.  Topology Appl., 125(2):263–278, 2002.
  * [10] R. Engelking.  General topology.  Heldermann, Berlin, 1989.  Revised and completed edition.
  * [11] D. Gauld.  Non-metrisable manifolds.  Springer-Verlag, New York-Berlin, 2014.
  * [12] K.P. Hart.  More remarks on Souslin properties and tree topologies.  Topology Appl., 15:151–158, 1983.
  * [13] A. Hatcher.  Algebraic topology.  Cambridge University Press, Cambridge, 2002.  (A free electronic version is available).
  * [14] M. Ismail and P. Nyikos.  On spaces in which countably compact sets are closed, and hereditary properties.  Topology Appl., 11(3):281–292, 1980.
  * [15] W. Just and M. Weese.  Discovering Modern Set Theory. II: Set-theoretic, tools for every mathematician, volume 18 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, 1997.
  * [16] K. Kunen.  Set theory, an introduction to independance proofs (reprint).  Number 102 in Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1983.
  * [17] A. Mardani.  Topics in the general topology of non-metric manifolds.  The University of Auckland. Ph.D Thesis, 2014.
  * [18] S. Mrówka.  Some set-theoretic constructions in topology.  Fund. Math., 94:83–92, 1977.
  * [19] P. Nyikos.  Antidiamond and anti-pfa axioms and topological applications.  Preliminary Draft on his webpage.
  * [20] P. Nyikos.  The theory of nonmetrizable manifolds.  In Handbook of Set-Theoretic Topology (Kenneth Kunen and Jerry E. Vaughan, eds.), pages 633–684. North-Holland, Amsterdam, 1984.
  * [21] P. Nyikos.  Progress on countably compact spaces.  In Z. Frolik, editor, General Topology and its Relations to Modern Analysis and Algebra VI, pages 379–406, Berlin, 1988. Heldermann Verlag.
  * [22] P. Nyikos.  On first countable, countably compact spaces III: The problem of obtaining separable non-compact examples.  In Open Problems in Topology, pages 127–161. North-Holland, Amsterdam, 1990.
  * [23] P. Nyikos.  Hereditary normality versus countable tightness in countably compact spaces.  Topology Appl., 44:271–292, 1992.
  * [24] P. Nyikos.  Various smoothings of the long line and their tangent bundles.  Adv. Math., 93(2):129–213, 1992.
  * [25] P. Nyikos.  Various topologies on trees.  In P.R. Misra and M. Rajagopalan, editors, Proceedings of the Tennessee Topology Conference, pages 167–198. World Scientific, 1997.
  * [26] A.J. Ostaszewski.  On countably compact, perfectly normal spaces.  J. London Math. Soc., 14:505–516, 1976.
  * [27] J.R. Porter and R.G. Woods.  Feebly compact spaces, Martin’s Axiom and “Diamond”.  Top. Proc., 9:105–121, 1984.