# Autohomeomorphisms of pre-images of $\mathbb{N}^{*}$

Alan Dow  Department of Mathematics and Statistics, UNC Charlotte

###### Abstract.

In the study of the Stone-C̆ech remainder of the real line a detailed study of
the Stone-C̆ech remainder of the space $\mathbb{N}\times[0,1]$, which we
denote as $\mathbb{M}$, has often been utilized. Of course the real line can
be covered by two closed sets that are each homeomorphic to $\mathbb{M}$. It
is known that an autohomeomorphism of $\mathbb{M}^{*}$ induces an
autohomeomorphism of $\mathbb{N}^{*}$. We prove that it is consistent with
there being non-trivial autohomeomorphism of $\mathbb{N}^{*}$ that those
induced by autohomeomorphisms of $\mathbb{M}^{*}$ are trivial.

###### Key words and phrases:

Stone-Cech, PFA, automorphisms

###### 1991 Mathematics Subject Classification:

03A35, 54D35, 54A35

##  1\. Introduction

We consider the remainder
$\mathbb{M}^{*}=\beta(\mathbb{N}\times[0,1])\setminus(\mathbb{N}\times[0,1])=\beta\mathbb{M}\setminus\mathbb{M}$
and the projection
$\pi:\mathbb{M}^{*}\rightarrow\mathbb{N}^{*}$ satisfying that
$\pi((a\times[0,1])^{*})=a^{*}$ for all infinite $a\subset\mathbb{N}$.

For each $u\in\mathbb{N}^{*}$, we let $\mathbb{I}_{u}$ denote the pre-image
$\pi^{-1}(u)$ of $u$. Each $\mathbb{I}_{u}$ is a continuum, and it is an
example of a standard subcontinuum as per [Hart92]. Perhaps it is appropriate
to refer to each of these (more special) $\mathbb{I}_{u}$ as primary standard
subcontinuum.

It was shown in [DH1993] that every autohomeomorphism $\Psi$ of
$\mathbb{M}^{*}$ induces an autohomeomorphism $H_{\Psi}$ of $\mathbb{N}^{*}$
satisfying that $\Psi(\mathbb{I}_{u})=\mathbb{I}_{H_{\Psi}(u)}$ (i.e.
$H_{\Psi}\circ\pi=\pi\circ\Psi$). The question addressed in this article,
raised in [DH1993], is whether every autohomeomorphism $H$ of $\mathbb{N}^{*}$
is induced by some autohomeomorphism of $\mathbb{M}^{*}$ in this manner. It is
evident that this question has an affirmative answer if all autohomemorphisms
of $\mathbb{N}^{*}$ are trivial. We prove in Theorem 5.1 below that it is
consistent that a negative answer is also consistent. The affirmative answer
under CH will appear in another paper. The main reference for results in this
section is [Hart92].

Within any compact space $X$ and a sequence $\\{A_{n}:n\in\mathbb{N}\\}$ of
subsets, it is common to let $u{-}\lim\\{A_{n}\\}_{n}$ denote the usual set of
$u$-limits, $\bigcap\\{\overline{\left(\bigcup_{n\in a}A_{n}\right)}:a\in
u\\}$.

In particular, if $\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ is a sequence of
pairwise disjoint connected subintervals of $\mathbb{N}\times I$, we let
$[a_{m},b_{m}]_{u}$ denote $u{-}\lim\\{[a_{m},b_{m}]\\}_{m}$. If there is some
$U\in u$ satisfying that the sequence $\\{[a_{m},b_{m}]:m\in U\\}$ is locally
finite in $\mathbb{N}\times I$, then $[a_{m},b_{m}]_{u}$ is also an example of
a standard subcontinuum of $\mathbb{M}^{*}$.

Say that a sequence $\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ is a standard
sequence if it is a locally finite set of pairwise disjoint non-trivial
connected intervals in $\mathbb{N}\times I$. A standard sequence will be
called rational if all the end-points are rational numbers.

######  Definition 1.1.

If $\mathcal{A}=\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ is a standard sequence,
let $\mathbb{M}^{*}_{\mathcal{A}}$ denote the set
$\mbox{cl}_{\beta\mathbb{M}}(\bigcup\mathcal{A})\setminus\mathbb{M}$.

######  Lemma 1.2.

If $\mathcal{A}$ is a standard sequence, then $\bigcup\mathcal{A}$ is
homeomorphic to $\mathbb{M}$, and $\mathbb{M}^{*}_{\mathcal{A}}$ is
homeomorphic to $\mathbb{M}^{*}$.

######  Proposition 1.3.

If $K$ and $L$ are disjoint compact subsets of $\mathbb{M}^{*}$, then there
are standard sequences $\mathcal{A}=\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ and
$\mathcal{C}=\\{[c_{m},d_{m}]:m\in\mathbb{N}\\}$ and disjoint sets
$N_{K},N_{L}$ of $\mathbb{N}$ such that

  1. (1)

for all $m\in\mathbb{N}$, $a_{m}<c_{m}<d_{m}<b_{m}$,

  2. (2)

$K$ is contained in the $\beta\mathbb{M}$-closure of
$\bigcup\\{[c_{m},d_{m}]:m\in N_{K}\\}$

  3. (3)

$L$ is contained in the $\beta\mathbb{M}$-closure of
$\bigcup\\{[c_{m},d_{m}]:m\in N_{L}\\}$.

A point $x$ of $[a_{m},b_{m}]_{u}$ is a cut-point if for every
$f\in\mathbb{N}^{\mathbb{N}}$, there is a sequence $a_{m}<c_{m}<d_{m}<b_{m}$
($m\in\mathbb{N}$) such that

  1. (1)

$x$ is a cut-point of $[c_{m},d_{m}]_{u}$,

  2. (2)

$\\{m\in\mathbb{N}:d_{m}-c_{m}<1/f(m)\\}\in u$.

Say that a standard sequence $\\{[c_{m},d_{m}]:m\in\mathbb{N}\\}$ is $f$-thin
if $d_{m}-c_{m}<1/f(m)$ for all $m\in\mathbb{N}$.

Certainly, if $a_{m}<x_{m}<b_{m}$ (for all $m\in\mathbb{N}$), then
$x_{u}=u{-}\lim\\{x_{m}:m\in\mathbb{N}\\}$ is a (standard) cut-point of
$[a_{m},b_{m}]_{u}$.

The standard cut-points of $[a_{m},b_{m}]_{u}$ are naturally linearly ordered
as in the ultraproduct. The closure of any subinterval of the standard cut-
points is said to be a subinterval of $[a_{m},b_{m}]_{u}$.

If $\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ is a standard sequence then for every
selection $a_{m}\leq x_{m}\leq b_{m}$, the set
$\\{x_{u}:u\in\mathbb{N}^{*}\\}$ maps homeomorphically to $\mathbb{N}^{*}$ by
the map $x_{u}$ being sent to $u$. Say that a sequence
$\\{x_{m}:m\in\mathbb{N}\\}$ is a selector sequence (for
$\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$) if $a_{m}<x_{m}<b_{m}$ for all
$m\in\mathbb{N}$.

######  Proposition 1.4.

If a subset $L$ of $\mathbb{M}^{*}$ is homeomorphic to an interval of any
standard subcontinuum of any standard sequence then it is an actual
subinterval of a standard subinterval of a standard sequence.

I believe this next result is new. For any continuous $g:\mathbb{M}\rightarrow
I$ let $g^{*}$ denote the natural extension of $g$ mapping $\mathbb{M}^{*}$ to
$I$.

######  Theorem 1.5.

Suppose that $\Psi:\mathbb{M}^{*}\rightarrow\mathbb{M}^{*}$ is a homeomorphism
and let $\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ be a standard sequence in
$\mathbb{M}$. Then, for any selector sequence $\\{x_{m}:m\in\mathbb{N}\\}$ for
$\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$, there is a standard sequence
$\\{[c_{j},d_{j}]:j\in\mathbb{N}\\}$ and a homeomorphism
$h:\mathbb{N}^{*}\rightarrow\mathbb{N}^{*}$ such that for each
$u\in\mathbb{N}^{*}$,

  1. (1)

$\Psi(x_{u})\in[c_{j},d_{j}]_{h(u)}$,

  2. (2)

$[c_{j},d_{j}]_{h(u)}$ is a subinterval of $\Psi([a_{m},b_{m}]_{u})$,

  3. (3)

the mapping $\Psi(x_{u})$ to $h(u)$ is a homeomorphism from
$\Psi(\\{x_{m}:m\in\mathbb{N}\\}^{*})=\\{\Psi(x_{u}):u\in\mathbb{N}^{*}\\}$ to
$\mathbb{N}^{*}$.

###### Proof.

Choose a continuous function $g:\mathbb{M}\rightarrow I$ satisfying that, for
every $m\in\mathbb{N}$, $g(x_{m})=1$ and $g(r)=0$ for all
$r\notin\bigcup\\{(a_{m},b_{m}):m\in\mathbb{N}\\}$.

Choose a continuous function $g_{1}:\mathbb{M}\rightarrow I$ satisfying that
$g_{1}^{*}=g^{*}\circ\Psi^{-1}$. For each $u\in\mathbb{N}^{*}$, let
$y_{u}=\Psi(x_{u})$. Observe that $g_{1}^{*}(y_{u})=1$ for all
$u\in\mathbb{N}^{*}$.

Let $\\{[c_{j},d_{j}]:j\in\mathbb{N}\\}$ enumerate all maximal connected
intervals in $\mathbb{M}$ that satisfy that $(c_{j},d_{j})\cap
g_{1}^{-1}(\frac{1}{2},1]$ is dense in $[c_{j},d_{j}]$.

Consider any $u\in\mathbb{N}^{*}$ and the subcontinuum
$\tilde{I}_{u}=\Psi(I_{u})$. There is a standard interval subsequence
$\\{[\tilde{a}_{m},\tilde{b}_{m}]:m\in\mathbb{N}\\}$ of
$\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ that has $\\{x_{m}:m\in\mathbb{N}\\}$ as
a selector and is contained in $g^{-1}(\frac{3}{4},1])$. The point $y_{u}$ is
in $\Psi([\tilde{a}_{m},\tilde{b}_{m}]_{u})$ which is a subinterval of
$\tilde{I}_{u}$. Also $\Psi([\tilde{a}_{m},\tilde{b}_{m}]_{u})$ is contained
in the interior of $g_{1}^{-1}(\frac{1}{2},1]$. Now choose any standard
sequence $\\{[r_{n},s_{n}]:n\in\mathbb{N}\\}$ so that
$g_{1}\left(\bigcup\\{[r_{n},s_{n}]:n\in\mathbb{N}\\}\right)^{*}\subset(\frac{1}{2},1]$
and so that $\Psi([\tilde{a}_{m},\tilde{b}_{m}]_{u})$ is a subinterval of
$[r_{n},s_{n}]_{v}$ for some (unique) ultrafilter $v\in\mathbb{N}^{*}$.

For each $n\in\mathbb{N}$, choose the unique $j_{n}\in\mathbb{N}$ so that
$[r_{n},s_{n}]\subset(c_{j_{n}},d_{j_{n}})$. Since
$\\{[r_{n},s_{n}]:n\in\mathbb{N}\\}$ is locally finite, the sequence
$\\{[c_{j_{n}},d_{j_{n}}]:n\in\mathbb{N}\\}$ is also locally finite. Hence
$\\{[c_{j_{n}},d_{j_{n}}]:n\in\mathbb{N}\\}$ is a standard sequence. The
mapping $n\mapsto j_{n}$ is finite-to-one and now let $w$ be the finite-to-one
image of $v$. Consider the standard subcontinuum $[c_{j},d_{j}]_{w}$. We check
that $[c_{j},d_{j}]_{w}$ is an interval in $\tilde{I}_{u}$, and that $h(u)=w$
is the map that satisfies the statement of the Lemma.

The continuum $[c_{j},d_{j}]_{w}$ is contained in the component
$\tilde{I}_{u}$ of
$\Psi\left(~{}\left(\bigcup\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}\right)^{*}\right)$
and contains the interval $\Psi([\tilde{a}_{m},\tilde{b}_{m}]_{u})$. It
follows from the results in [Hart92], that $[r_{n},s_{n}]_{v}$ is an interval
in $[c_{j},d_{j}]_{w}$. Choose any continuous function
$g_{2}:\mathbb{M}\rightarrow I$ satisfying that $g_{2}^{*}(y_{u})=1$ and
$g_{2}^{*}(\tilde{I}_{u}\setminus[r_{n},s_{n}]_{v})=0$.

Set $L=\\{n:g_{2}([a_{n},b_{n}])\setminus[0,\frac{1}{4}]\neq\emptyset\\}$ and
$\tilde{L}=\\{j_{n}:n\in L\\}$. Since $g^{*}_{2}(y_{u})=1$, it follows that
$L\in v$ and $\tilde{L}\in w$. Let the function sending each $n$ to $j_{n}$ be
denoted by $\rho$. Thus $\rho^{*}(v)=w$. For every
$\tilde{v}\in\mathbb{N}^{*}$ such that $\rho^{*}(\tilde{v})=w$, we have that
$[r_{n},s_{n}]_{\tilde{v}}$ is a subset of
$[c_{j},d_{j}]_{w}\subset\tilde{I}_{u}$. Therefore $v$ is the unique
ultrafilter satisfying that $\rho^{*}(v)=w$ and $L\in v$. Therefore, by
removing a finite set from $L$, we can assume that $\rho$ is 1-to-1 on $L$.

Next, we note that $y_{u}$ is in the interior of
$\left(\bigcup\\{[r_{n},s_{n}]:n\in L\right)^{*}$ (easily checked by the
results in [Hart92]) and so there is a $U\in u$ such that $\Psi(\\{x_{m}:m\in
U\\}^{*})\subset\left(\bigcup\\{[r_{n},s_{n}]:n\in L\right)^{*}$.

We now have our desired mapping defined on $U^{*}$. For each $\tilde{u}\in
U^{*}$, there is a unique $v_{\tilde{u}}\in L^{*}$ so that
$y_{\tilde{u}}\in[r_{n},s_{n}]_{v_{\tilde{u}}}$. The continuous projection
mapping on $\left(\bigcup\\{[r_{n},s_{n}]:n\in L\right)^{*}$ to $L^{*}$ agrees
with the mapping sending $y_{\tilde{u}}$ to $v_{\tilde{u}}$. The mapping
sending $\tilde{u}$ to $y_{\tilde{u}}$ is induced by the continuous mapping
$\Psi\restriction\\{x_{m}:m\in U\\}^{*}$. Therefore the mapping $h(\tilde{u})$
sent to $\rho^{*}(v_{\tilde{u}})$ is continuous (and 1-to-1).

∎

##  2\. $\mathbb{N}^{*}$ cut-sets

######  Definition 2.1.

Say that a subset $K$ of $\mathbb{M}^{*}$ is an $\mathbb{N}^{*}$ cut-set (for
the standard sequence) if there is a standard sequence
$\mathcal{A}=\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ such that

  1. (1)

$K$ is a subset $\mathbb{M}^{*}_{\mathcal{A}}$,

  2. (2)

$K\cap[a_{m},b_{m}]_{u}$ is a cut-point of $[a_{m},b_{m}]_{u}$ for every
$u\in\mathbb{N}^{*}$,

  3. (3)

the mapping from $\left(\bigcup\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}\right)^{*}$
where $[a_{m},b_{m}]$ is sent to $m$, to $\mathbb{N}^{*}$ is 1-to-1 on $K$.

An $\mathbb{N}^{*}$ cut-set $K$ is trivial if $K$ equals $D^{*}$ for some
closed discrete $D\subset\mathbb{M}$.

A concept called non-trivial maximal nice filters on spaces of the form
$\mathbb{N}\times X$ (with $X$ compact) was introduced and studied in
[DowPAMS] where it was shown that PFA implies these do not exist if $X$ is
metrizable. The neighborhood filters in $\beta\mathbb{M}$ of a
$\mathbb{N}^{*}$ cut-set result in maximal nice filters.

Let us a recall that a set $K$ is said to be a $P_{\kappa}$-set in a space $X$
if every $G_{<\kappa}$ of $X$ that contains $K$ has $K$ in its interior. It is
well-known that $D^{*}$ is a $P_{\mathfrak{b}}$-set for every closed subset
$D$ of $\mathbb{M}$. In this next result we prove that an $\mathbb{N}^{*}$
cut-set has a neighborhood base resembling that of a cut-point. This result
isn’t strictly needed since the $\mathbb{N}^{*}$ cut-sets that we intend to
consider are those of the form $\Psi(D^{*})$ when $D^{*}$ is a trivial
$\mathbb{N}^{*}$ cut-set.

######  Proposition 2.2.

Every $\mathbb{N}^{*}$ cut-set $K$ of $\mathbb{M}^{*}$ is a
$P_{\mathfrak{b}}$-set in $\mathbb{M}^{*}$. Also, if the standard sequence
$\mathcal{A}=\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ witnessing that $K$ is an
$\mathbb{N}^{*}$ cut-set, then there is a family of rational standard
sequences $\\{\mathcal{C}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$ such that, for
each $f\in\mathbb{N}^{\mathbb{N}}$,

$\mathcal{C}_{f}=\\{[c^{f}_{m},d^{f}_{m}]\subset[a_{m},b_{m}]:m\in\mathbb{N}\\}$
is $f$-thin

and the family
$\\{\left(\bigcup\mathcal{C}_{f}\right)^{*}:f\in\mathbb{N}^{\mathbb{N}}\\}$ is
a neighborhood base for $K$ in $\mathbb{M}_{\mathcal{A}}^{*}$. Note also that
the family
$\\{\left(\bigcup\mathcal{C}_{f}\right)^{*}:f\in\mathbb{N}^{\mathbb{N}}\\}$ is
${<}\mathfrak{b}$-directed.

###### Proof.

Let $\mathcal{A}=\\{[a_{m},b_{m}]:m\in\mathbb{N}\\}$ be a standard sequence
witnessing that $K$ is an $\mathbb{N}^{*}$ cut-set. Let $U$ be an open subset
of $\beta\mathbb{M}_{\mathcal{A}}$ that contains $K$. Select, by Lemma 1.3, a
pair $\tilde{\mathcal{A}}=\\{[\tilde{a}_{m},\tilde{b}_{m}]:m\in\mathbb{N}\\}$
and $\tilde{\mathcal{C}}=\\{[\tilde{c}_{m},\tilde{d}_{m}]:m\in\mathbb{N}\\}$
so that there are disjoint subsets $N_{K},N_{U}$ of $\mathbb{N}$ so that

  1. (1)

for all $m\in\mathbb{N}$,
$\tilde{a}_{m}<\tilde{c}_{m}<\tilde{d}_{m}<\tilde{b}_{m}$,

  2. (2)

for all $m\in\mathbb{N}$ there is a $k$ such that
$[\tilde{a}_{m},\tilde{b}_{m}]\subset[a_{k},b_{k}]$,

  3. (3)

$K$ is contained in the $\beta\mathbb{M}$-closure of
$\bigcup\\{[\tilde{c}_{m},\tilde{d}_{m}]:m\in N_{K}\\}$,

  4. (4)

$\mathbb{M}_{\mathcal{A}}^{*}\setminus U_{n}$ is contained in the
$\beta\mathbb{M}$-closure of $\bigcup\\{[\tilde{c}_{m},\tilde{d}_{m}]:m\in
N_{U_{n}}\\}$.

Let $\psi$ denote the function on $\mathbb{M}_{\mathcal{A}}$ onto $\mathbb{N}$
obtained by sending every point of $[a_{m},b_{m}]$ to $m$. We may let
$\psi^{*}$ then denote the continuous extension of $\psi$ to
$\mathbb{M}_{\mathcal{A}}^{*}$ onto $\mathbb{N}^{*}$.

For each $x\in K$ (a cut-point) with $\psi^{*}(x)=u\in\mathbb{N}^{*}$, there
is a set $L_{x}\subset N_{K}$ such that $\psi\restriction L_{x}$ is 1-to-1 and
$x$ is in the $\beta\mathbb{M}$-closure of the union of the standard sequence
$\tilde{\mathcal{A}}_{L_{x}}=\\{[\tilde{a}_{m},\tilde{b}_{m}]:m\in L_{x}\\}$.
The closure of the $\tilde{\mathcal{A}}_{L_{x}}$ meets $K$ in a clopen set and
so, by possibly shrinking $L_{x}$ we can ensure that $x$ is still in the
closure and that this clopen subset of $K$ is equal to the intersection of $K$
with the closure of the union of the set $\\{[a_{j},b_{j}]:j\in\psi[L_{x}]\\}$
(i.e. all the $\mathcal{A}$-intervals that are hit by
$\tilde{\mathcal{A}}_{L_{x}}$). Since there is a finite cover of $K$ by such
clopen sets (associated with the closures of these
$\tilde{\mathcal{A}}_{L_{x}}$’s), this shows that there is a selector function
$\sigma:\mathbb{N}\rightarrow\mathbb{N}$ so that

  1. (1)

for each $m\in\mathbb{N}$, $[\tilde{a}_{\sigma(m)},\tilde{b}_{\sigma(m)}]$ is
a subset of $[a_{m},b_{m}]$,

  2. (2)

the closure of the union of the intervals in
$\\{[\tilde{a}_{\sigma(m)},\tilde{b}_{\sigma(m)}]:m\in\mathbb{N}\\}$ contains
$K$ in its interior,

  3. (3)

the closure of the union of the intervals in
$\\{[\tilde{a}_{\sigma(m)},\tilde{b}_{\sigma(m)}]:m\in\mathbb{N}\\}$ is
contained in $U_{n}$.

Choose any function $f\in\mathbb{N}^{\mathbb{N}}$ so that for all but finitely
many $m$, each of the distances
$\\{|\tilde{a}_{\sigma(m)}-\tilde{c}_{\sigma(m)}|,|\tilde{d}_{\sigma(m)}-\tilde{c}_{\sigma(m)}|,\\{|\tilde{d}_{\sigma(m)}-\tilde{b}_{\sigma(m)}|\\}$
are greater than $5/f(m)$. For every $x\in K$, applying the definition of cut-
point, choose an $f$-thin standard sequence
$\mathcal{C}_{x}=\\{[c^{x}_{m},d^{x}_{m}]:m\in\mathbb{N}\\}$ so that
$x\in[c^{x}_{m},d^{x}_{m}]_{u}$ where $u=\psi^{*}(x)$. Again, we can choose
$L_{x}\subset\mathbb{N}$ so that $L_{x}\in u$ and so that for all $y\in K$
with $L_{x}\in\psi^{*}(y)$, we also have that $y$ is in the $\beta\mathbb{M}$
closure of the union of $\mathcal{C}_{x}$. In fact $L_{x}$ is simply any set
such that
$L_{x}^{*}=\psi^{*}(K\cap\mbox{cl}_{\beta\mathbb{M}}\left(\bigcup\\{[c^{x}_{m},d^{x}_{m}]:m\in\mathbb{N}\\}\right))$.
If needed, we can make ${<}1/f(n)$ changes to any of the $c^{x}_{m}$’s and
$d^{x}_{m}$’s so as to ensure every $y\in K$ with $L_{x}\in\psi^{*}(y)$, is in
the interior of
$\mbox{cl}_{\beta\mathbb{M}}\left(\bigcup\\{[c^{x}_{m},d^{x}_{m}]:m\in\mathbb{N}\\}\right)$.
Now consider the sequence
$\\{[\tilde{a}_{\sigma(m)},\tilde{b}_{\sigma(m)}]:m\in\mathbb{N}\\}$. It
follows that, for all but finitely many $m\in L_{x}$, the interval
$[c^{x}_{m},d^{x}_{m}]$ meets the interval
$[\tilde{c}_{\sigma(m)},\tilde{d}_{\sigma(m)}]$. By the choice of $f$, we then
have that, for all but finitely many $m\in L_{x}$, the interval
$[c^{x}_{m},d^{x}_{m}]$ is contained in
$[\tilde{a}_{\sigma(m)},\tilde{b}_{\sigma(m)}]$. This shows that
$\left(\bigcup\\{[c^{x}_{m},d^{x}_{m}]:m\in\mathbb{N}\\}\right)^{*}$ is
contained in $U$ and is a neighborhood of a relatively clopen set of points of
$K$. By the compactness of $K$ there is an $f$-thin standard sequence
$\mathcal{C}_{f}=\\{[c^{f}_{m},d^{f}_{m}]\subset[a_{m},b_{m}]:m\in\mathbb{N}\\}$
such that $\left(\bigcup\mathcal{C}_{f}\right)^{*}$ is contained in $U$.

Now we prove that the family is ${<}\mathfrak{b}$-directed. Now let
$F\subset\mathbb{N}^{\mathbb{N}}$ be any set with $|F|<\mathfrak{b}$. For each
$f\in F$, choose also $g_{f}\in\mathbb{N}^{\mathbb{N}}$ so that $f<g_{f}$ and
for all $m\in\mathbb{N}$,
$[c^{m}_{g_{f}},d^{m}_{g_{f}}]\subset(c^{m}_{f},d^{m}_{f})$ ( using that the
closure of $D=\\{c^{m}_{f},d^{m}_{f}:m\in\mathbb{N}\\}$ in $\beta\mathbb{M}$
is disjoint from $K$. For each $f\in F$, let $h_{f}\in\mathbb{N}^{\mathbb{N}}$
satisfy that, for all $m\in\mathbb{N}$, $5/h_{f}(m)$ is less than each of the
values $c^{m}_{g_{f}}-c^{m}_{f}$, $d^{m}_{f}-d^{m}_{g_{f}}$, and
$d^{m}_{g_{f}}-c^{m}_{g_{f}}$. Choose $\bar{f}\in\mathbb{N}^{\mathbb{N}}$ so
that for all $f\in F$, $h_{f}<^{*}\bar{f}$. Fix any $f\in F$, and note that
there is some $m_{f}$ so that, for all $m>m_{f}$, we have that
$[c^{m}_{\bar{f}},d^{m}_{\bar{f}}]\cap[c^{m}_{g_{f}},d^{m}_{g_{f}}]$ is not
empty and that $h_{f}(m)<f(m)$. Consider any $m>m_{f}$ and the required
condition: $c^{m}_{g_{f}}\leq d^{m}_{\bar{f}}$ or $c^{m}_{\bar{f}}\leq
d^{m}_{g_{f}}$. In the first case we then have

$c^{m}_{f}<c^{m}_{g_{f}}-\frac{1}{\bar{f}(m)}<d^{m}_{\bar{f}}-\frac{1}{\bar{f}(m)}<c^{m}_{\bar{f}}<d^{m}_{\bar{f}}<c^{m}_{\bar{f}}+\frac{1}{\bar{f}(m)}<c^{m}_{g_{f}}+\frac{1}{g_{f}(m)}<d^{m}_{g_{f}}<d^{m}_{f}$

which implies that
$[c^{m}_{\bar{f}},d^{m}_{\bar{f}}]\subset(c^{m}_{f},d^{m}_{f})$. The case when
$c^{m}_{f}\leq d^{m}_{g_{f}}$ also implies that
$[c^{m}_{\bar{f}},d^{m}_{\bar{f}}]\subset(c^{m}_{f},d^{m}_{f})$ by a similar
argument. ∎

######  Definition 2.3.

Say that an indexed family $\\{\mathcal{C}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$
is an $\mathbb{N}^{*}$ cut-set of $\mathbb{M}^{*}$

  1. (1)

every $\mathcal{C}_{f}$ is a rational standard sequence indexed as
$\\{[c^{m}_{f},d^{m}_{f}]:m\in\mathbb{N}\\}$,

  2. (2)

the sequence $\mathcal{C}_{f}$ is $f$-thin,

  3. (3)

the family is countably directed in the sense that, for each family
$\\{f_{n}:n\in\omega\\}\subset\mathbb{N}^{\mathbb{N}}$, there is an
$f\in\mathbb{N}$ such that, for all $n\in\omega$, $f<^{*}f_{n}$ and
$\\{m:[c^{m}_{f},d^{m}_{f}]\not\subset(c^{m}_{f_{n}},d^{m}_{f_{n}})\\}$ is
finite.

Given any $\mathcal{C}_{f}$ and subset $I$ of $\mathbb{N}$, let
$\mathcal{C}_{f}\restriction I=\\{[c^{m}_{f},d^{m}_{f}]:m\in I\\}$. For an
$\mathbb{N}^{*}$ cut-set family
$\mathfrak{C}=\\{\mathcal{C}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$, and a subset
$I\subset\mathbb{N}$, let $\mathfrak{C}\restriction I$ denote the (re-indexed)
$\mathbb{N}^{*}$ cut-set $\\{\mathcal{C}_{f}\restriction
I:f\in\mathbb{N}^{\mathbb{N}}\\}$. Let $\mbox{Triv}(\mathfrak{C})$ denote the
ideal of subsets $I$ of $\mathbb{N}$ satisfying that the family
$\mathfrak{C}\restriction I$ is trivial.

##  3\. The effects of two consequences of PFA

In this section we work with two assumptions $(\dagger_{1}^{+})$ and
$(\dagger_{2}^{+})$, each of which is a consequence of PFA. In fact,
$(\dagger_{2}^{+})$ is a consequence of the open graph axiom (formerly OCA).
Let $H$ be an autohomeomorphism of $\mathbb{N}^{*}$. For a subset $a$ of
$\mathbb{N}$, $H$ is said to be trivial on $a^{*}$ if (by possibly removing a
finite subset of $a$) there is a bijection $h_{a}$ from $a$ into $\mathbb{N}$
such that $H(c^{*})=\left(h_{a}(c)\right)^{*}$ for all $c\subset a$. The
family, $\mbox{Triv}(H)=\\{a\subset\mathbb{N}:H\ \mbox{is trivial on }\
a^{*}\\}$, is an ideal on $\mathcal{P}(\mathbb{N})$. An ideal $\mathcal{I}$ on
a countable set $D$ is said to be ccc over fin, if given any uncountable
family of pairwise almost disjoint subsets of $D$, all but countably many of
them are in $\mathcal{I}$.

Thus one says that an autohomeomorphism $H$ of $\mathbb{N}^{*}$ (or of $D^{*}$
for any countable discrete set $D$) is trivial modulo ccc over fin, if the
ideal $\mbox{Triv}(H)$ is ccc over fin. Similarly $H$ is somewhere trivial if
there is some infinite set $I\in\mbox{Triv}(H)$.

######  Definition 3.1.

The statement $(\dagger_{1}^{+})$ is the assertion that every
autohomeomorphism of $\mathbb{N}^{*}$ is trivial. The statement
$(\dagger_{1})$ is the statement that every such autohomeomorphism is trivial
modulo ccc over fin. While we are at it, let $(\dagger_{1}^{-})$ be the
statement that every autohomeomorphism of $\mathbb{N}^{*}$ is somewhere
trivial.

The statement $(\dagger_{2}^{+})$ is the assertion that every $\mathbb{N}^{*}$
cut-set of $\mathbb{M}^{*}$ is trivial, and again $(\dagger_{2})$ is the
statement that every $\mathbb{N}^{*}$ cut-set of $\mathbb{M}^{*}$ is trivial
on an ideal that is ccc over fin.

######  Lemma 3.2.

$(\dagger_{2}^{+})$ holds if $\mathfrak{c}=\aleph_{2}$ and the open graph
axiom holds.

###### Proof.

Let $\\{\mathcal{C}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$ be an $\mathbb{N}^{*}$
cut-set of $\mathbb{M}^{*}$. Using that the open graph axiom implies that
$\mathfrak{b}=\aleph_{2}$ and we are assuming $\mathfrak{c}=\aleph_{2}$, we
can recursively choose a dominating family
$\\{f_{\gamma}:\gamma\in\omega_{2}\\}\subset\mathbb{N}^{\mathbb{N}}$ so that,
by possibly making finite modifcations to each, the family
$\\{\mathcal{C}_{f_{\gamma}}:\gamma\in\omega_{2}\\}$ is directed as in
Proposition 2.2 and is cofinal in the family
$\\{\mathcal{C}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$. Let $F$ denote the family
$\\{f_{\gamma}:\gamma\in\omega_{2}\\}$ without the indexing.

Let $R$ be the set of pairs $(f,f^{\prime})\in[F]^{2}$ such that there is an
$m\in\mathbb{N}$ such that
$[c^{m}_{f},d^{m}_{f}]\cap[c^{m}_{f^{\prime}},d^{m}_{f^{\prime}}]$ is empty.
If we identify each $f\in F$ with the corresponding sequence $\mathcal{C}_{f}$
as an element of $\left(\mathbb{Q}^{2}\right)^{\mathbb{N}}$ viewed as a
product of discrete spaces, then the relation $R$ is open in the resulting
metric space. Assume that $\\{f_{\alpha}:\alpha\in\omega_{1}\\}\subset F$
satisfies that $(f_{\alpha},f_{\beta})\in R$ for all
$\alpha\neq\beta\in\omega_{1}$. We may choose an $f\in F$ so that, for all
$\alpha<\omega_{1}$, there is an $m_{\alpha}\in\mathbb{N}$ so that
$[c^{m}_{f},d^{m}_{f}]\subset(c^{m}_{f_{\alpha}},d^{m}_{f_{\alpha}})$ for all
$m>m_{\alpha}$. Fix an $\bar{m}\in\mathbb{N}$ and an uncountable
$\Lambda\subset\omega_{1}$ so that $m_{\alpha}=\bar{m}$ for all
$\alpha\in\Lambda$. Also choose uncountable $\Lambda_{1}\subset\Lambda$ so
that, for all $\alpha,\beta\in\Lambda_{1}$ and $m\leq\bar{m}$,
$[c^{m}_{f_{\alpha}},d^{m}_{f_{\alpha}}]=[c^{m}_{f_{\beta}},d^{m}_{f_{\beta}}]$.
We now have a contradiction since, for all $\alpha\neq\beta\in\Lambda_{1}$,
$[c^{m}_{f_{\alpha}},d^{m}_{f_{\alpha}}]\cap[c^{m}_{f_{\beta}},d^{m}_{f_{\beta}}]$
is not empty (contradicting the pair $(f_{\alpha},f_{\beta})$ is supposed to
be in $R$).

By the OGA, it follows there must be a cover of $F$ by a countable collection
$\\{F_{n}:n\in\omega\\}$ satisfying that $[F_{n}]^{2}$ is disjoint from $R$
for all $n\in\omega$. Choose any $n$ so that $F_{n}$ is a $<^{*}$-dominating
subfamily of $\mathbb{N}^{\mathbb{N}}$. By our construction, this also ensures
that $\\{\mathcal{C}_{f}:f\in F_{n}\\}$ is cofinal in the original family
$\\{\mathcal{C}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$ in the sense of
Proposition 2.2. Fix any $m\in\mathbb{N}$ and observe that the family
$\\{[c^{m}_{f},d^{m}_{f}]:f\in F_{n}\\}$ is linked. Of course this means we
can choose a sequence $\\{x_{m}:m\in\mathbb{N}\\}$ satisfying that
$x_{m}\in[c^{m}_{f},d^{m}_{f}]$ for all $f\in F_{n}$ and this contradicts that
there should be some $f\in F_{n}$ satisfying that the
$\beta\mathbb{M}$-closure of $\\{x_{m}\\}_{m\in\mathbb{N}}$ should be disjoint
from the $\beta\mathbb{M}$-closure of $\bigcup\mathcal{C}_{f}$.

∎

######  Theorem 3.3 (Assume $(\dagger_{1})$ and $(\dagger_{2})$).

Every autohomeomorphism $\Psi$ on $\mathbb{M}^{*}$ induces a trivial
autohomeomorphism on $\mathbb{N}^{*}$.

###### Proof.

Assume that $\Psi$ is an autohomeomorphism of $\mathbb{M}$. For simplicity
assume that
$\Psi(\,(\mathbb{N}\times\\{0\\})^{*}\,)=(\mathbb{N}\times\\{0\\})^{*}$. Let
$H$ denote the autohomeomorphism on $\mathbb{N}^{*}$ induced by $\Psi$, hence
$H\circ\pi=\pi\circ\Psi$.

Start with the continuous function $g:\mathbb{M}\rightarrow[0,1]$ satisfying
that, for each $n\in\mathbb{N}$,

  1. (1)

$g^{-1}(0)\cap(\\{n\\}\times[0,1])=\\{n\\}\times\\{\frac{i}{n}:0\leq i\leq
n\\}$,

  2. (2)

$g^{-1}(1)\cap(\\{n\\}\times[0,1])=\\{n\\}\times\\{\frac{2i+1}{2n}:0\leq
i<n\\}$,

  3. (3)

$g\restriction\left(\\{n\\}\times[\frac{i}{2n},\frac{i+1}{2n}]\right)$ is
linear for all $0\leq i<2n$.

Simply the graph of $g\restriction(\\{n\\}\times[0,1])$ oscillates $0$ to $1$
and back to $0$ exactly $n$ times. Let
$D=\\{(n,q):n\in\mathbb{N},q\in\\{\frac{i}{2n}:0<i<2n\\}$. Of course $D$ is a
closed discrete subset of $\mathbb{M}$. For an infinite subset $J$ of
$\mathbb{N}$, let $D_{J}=D\cap\left(J\times[0,1]\right)$.

Let $g_{1}$ be a continuous map on $\mathbb{M}$ satisfying that
$g_{1}^{*}=g^{*}\circ\Psi^{-1}$. For each $n\in\mathbb{N}$, let us count, call
it $L_{n}$, the number of times that $g_{1}$ oscillates as follows. Fix any
$n\in\mathbb{N}$ and set $t^{n}_{0}=0$ and define (if it exists)
$t^{n}_{1}=\inf\\{r:g_{1}(n,r)=\frac{2}{3}\\}$ (we think of $g_{1}$ as “on its
way towards 1”). Next, we define $t^{n}_{2}$ to be
$\inf\\{r\in[t^{n}_{1},1]:g_{1}(n,r)=\frac{1}{3}\\}$ (i.e. “$g_{1}$ is on its
way back towards 0”). Continue recursively defining this increasing sequence
$t^{n}_{0},t^{n}_{1},\ldots,t^{n}_{L_{n}}$ so that
$g_{1}(n,t^{n}_{i})=\frac{1}{3}$ for even $i\leq L_{n}$ and
$g_{1}(n,t^{n}_{i})=\frac{2}{3}$ for odd $i\leq L_{n}$. Also,
$g_{1}\restriction\left(\\{n\\}\times[t^{n}_{i},t^{n}_{i+1}]\right)$ is either
contained in $[0,\frac{2}{3}]$ or in $[\frac{1}{3},1]$. For each $0\leq
i<L_{n}$ also choose $s^{n}_{i}\in[t^{n}_{i},t^{n}_{i+1}]$ so that, if $i$ is
even, $g_{1}((n,s^{n}_{i}))=\min(g_{1}(\\{n\\}\times[t^{n}_{i},t^{n}_{i+1}])$,
and, if $i$ is odd,
$g_{1}((n,s^{n}_{i}))=\max(g_{1}(\\{n\\}\times[t^{n}_{i},t^{n}_{i+1}])$. It
follows from the fact that $g_{1}^{*}=g^{*}\circ\Psi$, that if $0\leq
i_{n}<L_{n}$ ($n\in\mathbb{N}$) is a sequence with each $i_{n}$ even, then
$\limsup\\{g_{1}((n,s^{n}_{i_{n}})):n\in\mathbb{N}\\}=0$. Similarly the limit
would be $1$ for a sequence of odd values for $i_{n}$. It also follows
similarly, that the sequence $\\{L_{n}:n\in\mathbb{N}\\}$ is, mod finite, a
diverging sequence of even numbers. For convenience (by possibly redefining
finitely many) we assume $L_{n}$ is even for all $n\in\mathbb{N}$.

Let $\sigma$ be the function in ${}^{\mathbb{N}}\,\mathbb{N}$ defined by
$\sigma(m)=L_{m}/2$ (it will help to change letters). Let us assume that $H$
is not trivial (meaning $\mathbb{N}$ is not in $\mbox{Triv}(H)$). We break the
rest of the proof into cases based on properties of $\sigma$. In each case we
prove that there is an infinite $a\in\mbox{Triv}(H)$ satisfying that
$L_{h_{a}(n)}\neq n$ for all $n\in a$. Appealing to symmetry we complete the
proof in the case that $L_{h_{a}(n)}<n$ for all $n\in a$. Note that
$(\dagger_{1})$ implies that every infinite $b\subset\mathbb{N}$ has an
infinite subset that is in $\mbox{Triv}(H)$.

First case is when $\sigma$ is, mod finite, a permutation on $\mathbb{N}$.
Since $\mathbb{N}\notin\mbox{Triv}(H)$, the almost permutation $\sigma^{-1}$
does not induce $H$. There is an infinite $b\subset\mathbb{N}$ such that
$\left(\sigma^{-1}(b)\right)^{*}$ and $H(b^{*})$ are distinct. By taking
complements if needed, we can assume that $\left(\sigma^{-1}(b)\right)^{*}$ is
not contained in $H(b^{*})$. Then choose an infinite $J\subset b$ so that
$\left(\sigma^{-1}(J)\right)^{*}$ is disjoint from $H(b^{*})\supset H(J^{*})$.
Now by $(\dagger_{1})$ we can assume that $J\in\mbox{Triv}(H)$ and choose the
injection $h_{J}:J\rightarrow\mathbb{N}$ witnessing that $a\in\mbox{Triv}(H)$.
It follows that, for every $n\in J$, $L_{h_{J}(n)}\neq n$.

Next case is that there is some infinite $b\subset\mathbb{N}$ such that $b$ is
disjoint from $\\{L_{m}:m\in\mathbb{N}\\}$. Again choose an infinite $J\subset
b$ in $\mbox{Triv}(H)$ and we again have $L_{h_{J}(n)}\neq n$ for all $n\in
J$.

The final case is that $\sigma$ is not 1-to-1. Choose any disjoint pair
$b_{0},b_{1}$ of infinite subset of $\mathbb{N}$ satisfying that $\sigma$ is
1-to-1 on each while $\sigma[b_{0}]=\sigma[b_{1}]$. First choose an infinite
$a_{0}\subset b_{0}$ with $a_{0}\in\mbox{Triv}(H)$. Next choose $a_{1}\subset
b_{1}$ so that $a_{1}\in\mbox{Triv}(H)$ and
$\sigma[a_{1}]\subset\sigma[a_{0}]$. Shrink $a_{0}$ so that
$\sigma[a_{0}]=\sigma[a_{1}]$. We may choose $J\in\mbox{Triv}(H)$ to be a
subset of one of $a_{0},a_{1}$ so that again $L_{h_{a}(n)}\neq n$ for all
$n\in a$.

Now we continue the proof by assuming that $J$ is an infinite set in
$\mbox{Triv}(H)$ satisfying that $L_{h_{J}(n)}<n$ for all $n\in J$.

Choose a standard sequence (indexed by $D$) $\\{[a_{d},b_{d}]:d\in D\\}$ so
that $d\in(a_{d},b_{d})$ for all $d\in D$. Of course $D$ is a selector set for
the standard sequence $\mathcal{A}=\\{[a_{d},b_{d}]:d\in D\\}$ and $D^{*}$ is
an $\mathbb{N}^{*}$ cut-set for $\mathcal{A}$. We pass to the subset
$D_{J}=D\cap\left(J\times[0,1]\right)$ and
$\mathcal{A}_{J}=\\{[a_{d},b_{d}]:d\in D_{J}\\}$. More generally, for
$I\subset J$, let $\mathcal{A}_{I}=\\{[a_{d},b_{d}]:d\in D_{I}\\}$.

By considering an uncountable almost disjoint family of infinite subsets of
$J$ and using Theorem 1.5 we have that $(\dagger_{2})$ implies that the
$\mathbb{N}^{*}$ cut-set $\Psi(D_{J}^{*})$ is trivial on an ideal that is ccc
over fin, there is an infinite $I\subset J$ so that $\Psi(D_{I}^{*})=K$ is a
trivial $\mathbb{N}^{*}$ cut-set with respect to the homeomorphic copy of
$\mathbb{M}^{*}$ we get from $\left(H(I)\times[0,1]\right)^{*}$. Therefore we
may choose a closed discrete $E\subset H(I)\times[0,1]$ so that $E^{*}=K$.
Notice that $\Psi\restriction D_{I}^{*}$ is an homeomorphism from $D_{I}^{*}$
to $E^{*}$. Again, by using $(\dagger_{1})$ and by possibly further shrinking
$I$ in the same manner that we shrunk $J$ to obtain $I$, we can assume that
$\Psi\restriction\left(D_{I}^{*}\cap\mathbb{M}_{I}^{*}\right)$ is trivial.
That means there is a (mod finite) bijection $f:D_{I}\rightarrow
E\cap\mathbb{M}_{H(I)}$ inducing $\Psi$ on $D_{I}^{*}\cap\mathbb{M}_{I}^{*}$.
We omit the easy verification that, by removing a finite set from $I$, we have
that for $d_{1}<d_{2}\in D\cap\left(\\{n\\}\times[0,1]\right)$ ($n\in I$),
$f(d_{1})<f(d_{2})\in E\cap\left(\\{h_{I}(n)\\}\times[0,1]\right)$.

Fix any $n\in I$, and for each $0\leq i\leq 2n$, let
$e_{i,n}=f((n,\frac{i}{2n}))$ (noting that $(n,\frac{i}{2n})\in D_{I}$).

######  Claim 1.

For all but finitely many $n\in b$,

  1. (1)

$g_{1}(e_{i,n})<\frac{1}{3}$ for all even $i\leq 2n$, and

  2. (2)

$g_{1}(e_{i,n})>\frac{2}{3}$ for all odd $i<2n$.

Indeed, $g^{*}$ will send every point of $\left(\\{(n,\frac{2i}{2n}):n\in
b,i\leq n\\}\right)^{*}$ to $0$, and so $g_{1}^{*}$ must send every point of
$\left(\\{f((n,\frac{2i}{2n}))=e_{2i,n}:n\in b,i\leq n\\}\right)^{*}$ to $0$.
The analogous property holds for the set of $e_{2i+1,n}$ ($n\in b$, $i<n$). It
thus follows that $g_{1}\restriction\left(h_{a}(n)\times[0,1]\right)$ must
oscillate at least $n$ times, contradicting that the oscillation number
$L_{h_{a}(n)}$ is less than $n$. ∎

##  4\. PFA

In this section, in working towards our main result, we give an alternate
proof of Vignati’s ([Vignati]) theorem that PFA implies that every
autohomeomorphism of $\mathbb{M}^{*}$ is trivial. We will work with
$(\dagger_{1}^{+})$ and $(\dagger_{2}^{+})$ and we also assume the principle
defined next.

######  Definition 4.1.

Say that $\mathcal{H}$ is an $\omega^{\omega}$-family if
$\mathcal{H}=\\{h_{f}:f\in\omega^{\omega}\\}$ is a family of functions
satisfying simply that
$\operatorname{dom}(h_{f})=\\{(n,m)\in\omega^{2}:m<f(n)\\}$. We then say that
such a family $\mathcal{H}$ is coherent, if whenever $f<^{*}g$ are in
$\omega^{\omega}$, then
$\\{(n,m)\in\operatorname{dom}(h_{f})\cap\operatorname{dom}(h_{g}):h_{f}((n,m))\neq
h_{g}((n,m))\\}$ is finite.

Say that the principle $\omega^{\omega}$-cohere holds if each
$\omega^{\omega}$-family $\mathcal{H}$ that is coherent, there is a function
$h$ with domain $\omega\times\omega$ such that $\mathcal{H}\cup\\{h\\}$ is
also coherent.

The principle $\omega^{\omega}$-cohere (not so named) is a well-known
consequence of OCA due to Todorcevic (see [Farah2000]*2.2.7). It is well-known
(essentially due to Hausdorff) that the principle $\omega^{\omega}$-cohere
implies that $\mathfrak{b}>\omega_{1}$. Todorcevic also proved the following
Proposition.

######  Proposition 4.2.

If $\omega_{1}<\mathfrak{b}$, then for any coherent $\omega^{\omega}$-family
$\mathcal{H}$, there is a countable set that mod finite contains the range of
each $h\in\mathcal{H}$.

###### Proof.

Assume there is no such countable set. Recursively choose
$\\{f_{\alpha}:\alpha<\omega_{1}\\}\subset\omega^{\omega}$, so that the range
of $h_{f_{\alpha}}$ is not, mod finite, contained in the union of the ranges
of the family $\\{h_{f_{\beta}}:\beta<\alpha\\}$. Choose any
$f\in\omega^{\omega}$ satisfying that $f_{\alpha}<^{*}f$ for all
$\alpha<\omega_{1}$. Let $L$ denote the range of $h_{f}$ and choose
$\delta<\omega_{1}$ large enough so that any point of $L$ that is in the range
of any $h_{f_{\alpha}}$ ($\alpha<\omega_{1}$) is already in the range of some
$h_{f_{\alpha}}$ with $\alpha<\delta$. We now have a contradiction to the
coherence assumption that guarantees that $h_{f_{\delta}}$ is, mod finite,
contained in $h_{f}$. ∎

######  Theorem 4.3.

If $(\dagger_{1}^{+}),(\dagger_{2}^{+})$ and the principle
$\omega^{\omega}$-cohere hold, then every autohomeomorphism of
$\mathbb{M}^{*}$ is trivial.

###### Proof.

Let $\Psi$ be an autohomeomorphism of $\mathbb{M}^{*}$. Since we are assume
$(\dagger_{1}^{+})$, there is no loss to assume that $\pi=\pi\circ\Psi$ (i.e.
$\Psi$ is a lifting of the identity map on $\mathbb{N}^{*}$). Fix an
enumeration, $\\{q_{\ell}:\ell\in\omega\\}$, of the rationals in $[0,1]$. For
every $f\in\omega^{\mathbb{N}}$, let
$D_{f}=\\{(n,q_{\ell}):n\in\mathbb{N},\ell<f(n)\\}$. Before proceeding we note
that the collection $\bigcup\\{(D_{f})^{*}:f\in\omega^{\mathbb{N}}\\}$ is a
dense subset of $\mathbb{M}^{*}$. By $(\dagger_{2})$, we may fix, for each
$f\in\omega^{\mathbb{N}}$, a countable set $E_{f}\subset\mathbb{M}$,
satisfying that $(E_{f})^{*}=\Psi((D_{f})^{*})$. Then by $(\dagger_{1}^{+})$,
we may fix a lifting $\sigma_{f}:D_{f}\rightarrow E_{f}$ that induces the
homeomorphism $\Psi\restriction(D_{f})^{*}$.

Loosely identifying $\mathbb{N}$ and $\\{q_{\ell}:\ell<\omega\\}$ with
$\omega$, it is apparent that
$\mathcal{H}=\\{\sigma_{f}:f\in\omega^{\mathbb{N}}\\}$ can be regarded as an
$\omega^{\omega}$-family. Since $\sigma_{f}$ and $\sigma_{g}$ (for $f<^{*}g$)
induce the same mapping on $(D_{f})^{*}$ (i.e. the mapping $\Psi$), it follows
that $\mathcal{H}$ is a coherent family. By the principle
$\omega^{\omega}$-cohere, we may choose a function
$\sigma:\mathbb{N}\times\\{q_{\ell}:\ell\in\omega\\}\rightarrow\mathbb{M}$
satisfying that $\sigma_{f}\subset^{*}\sigma$ for all
$f\in\omega^{\mathbb{N}}$. It follows that $\sigma$ is, mod finite, 1-to-1 and
$\sigma\restriction\left(\\{n\\}\times I\right)$ is an order-preserving
preserving function into $\\{n\\}\times I$ because any failure would violate
$\sigma_{f}\subset^{*}\sigma$ for some suitably large $f$. Similarly, it is
easily shown that the range of $\sigma$ is dense in $\\{n\\}\times I$ for all
but finitely many $n\in\mathbb{N}$.

The final thing to show is that, for all but finitely many $n\in\mathbb{N}$,
$\sigma\restriction\left(\\{n\\}\times[0,1]\right)$ is the restriction of a
homeomorphism $g_{n}:\\{n\\}\times[0,1]\rightarrow\\{n\\}\times[0,1]$, and
that the resulting (almost homeomorphism) $g=\bigcup_{n}g_{n}$ from
$\mathbb{M}$ to $\mathbb{M}$ satisfies that the Stone-C̆ech extension $\beta
g$ contains $\Psi$. By continuity and density, it suffices to show that for
any sequence $R=\\{(n,r_{n}):n\in\mathbb{N}\\}\subset\mathbb{M}$, for all but
finitely many $n\in\mathbb{N}$, $g_{n}$ is continuous at $(n,r_{n})$ and that
$\beta g\restriction R^{*}=\Psi\restriction R^{*}$. This we do now. Given such
a set $R$ (an $\mathbb{N}^{*}$ cut-set), we can choose a sequence
$S=\\{(n,s_{n}):n\in\mathbb{N}\\}\subset\mathbb{M}$ satisfying that
$\Psi(R^{*})=S^{*}$. Suppose there is an infinite set $a\subset\mathbb{N}$
satisfying that the oscillation of $\sigma$ on each neighborhood of
$(n,r_{n})$ is greater than some $\epsilon_{n}>0$. Apply the continuity of
$\Psi$ so as to choose a standard sequence $\\{\\{n\\}\times[c_{n},d_{n}]:n\in
a\\}$ satisfying that $c_{n}<r_{n}<d_{n}$ for all $n\in a$ and so that $\Psi$
sends the set $\left(\bigcup_{n\in a}\\{n\\}\times[c_{n},d_{n}]\right)^{*}$
into the set $\left(\bigcup_{n\in
a}\\{n\\}\times(s_{n}-\epsilon_{n}/4,s_{n}+\epsilon_{n}/4)\right)^{*}$. Next
choose two functions $\rho_{1},\rho_{2}\in\omega^{\mathbb{N}}$ satisfying
that, for all $n\in a$,
$\\{q_{\rho_{1}(n)},q_{\rho_{2}(n)}\\}\subset(c_{n},d_{n})$ and so that
$|\sigma((n,q_{\rho_{1}(n)}))-\sigma((n,q_{\rho_{2}(n)}))|>\epsilon_{n}$.
Choose any $f\in\omega^{\mathbb{N}}$ large enough so that
$\rho_{1}(n)+\rho_{2}(n)<f(n)$ for all $n\in\mathbb{N}$. Clearly there is an
infinite set $b\subset a$ such that, by symmetry,
$\sigma((n,q_{\rho_{1}(n)}))\notin[s_{n}-\epsilon_{n}/4,s_{n}+\epsilon_{n}/4]$
for all $n\in b$. But now the set $R_{1}=\\{(n,q_{\rho_{1}(n)}):n\in b\\}$ is
a subset of $R_{2}=D_{f}\cap\left(\bigcup_{n\in
a}\\{n\\}\times[c_{n},d_{n}]\right)$. Since $\Psi\restriction R_{2}^{*}$ is
induced by $\sigma$, we have a contradiction since $(\sigma(R_{1}))^{*}$ and
$\Psi(R_{2}^{*})$ are disjoint. ∎

##  5\. Non-trivial autohomeomorphisms

In this section we return to the question raised in [DH1993]: “Given an
autohomeomorphism $H:\mathbb{N}^{*}\rightarrow\mathbb{N}^{*}$, does there
exist an autohomeomorphism
$\Psi:\left(\mathbb{N}\times[0,1]\right)^{*}\rightarrow\left(\mathbb{N}\times[0,1]\right)^{*}$
satisfying $H\circ\pi=\pi\circ\Psi$?” We prove that it is consistent to have
$(\dagger_{1})$ and $(\dagger_{2})$ holding in a model in which there is a
non-trivial autohomeomorphism of $\mathbb{N}^{*}$ (i.e. $(\dagger_{1}^{+})$
fails). Once we succeed, then the following is a consequence of Theorem 3.3.

######  Theorem 5.1.

There is a model in which there are non-trivial autohomeomorphisms of
$\mathbb{N}^{*}$ and every automorphism on $\mathbb{N}^{*}$ induced by an
autohomeomorphism of $\mathbb{M}^{*}$ is trivial.

The model was introduced by Velickovic ([VelOCA]) and we will use the further
analysis of the forcing initiated in [ShStvel]. We do not know if
$(\dagger_{2}^{+})$ holds in this model. It would be interesting to have a
formulation of a suitable weakening of OCA that can be shown to hold in this
model.

First we define the partial order ${\mathbb{P}}_{2}$ from [VelOCA]

######  Definition 5.2.

The partial order ${\mathbb{P}}_{2}$ is defined to consist of all 1-to-1
functions $f$ where

  1. (1)

$\operatorname{dom}(f)=\operatorname{range}(f)\subset\mathbb{N}$,

  2. (2)

for all $i\in\operatorname{dom}(f)$ and $n\in\omega$, $f(i)\in[2^{n},2^{n+1})$
if and only if $i\in[2^{n},2^{n+1})$

  3. (3)

$\limsup_{n\rightarrow\omega}\lvert[2^{n},2^{n+1})\setminus\operatorname{dom}(f)\rvert=\omega$

  4. (4)

for all $i\in\operatorname{dom}(f)$, $i=f^{2}(i)\neq f(i)$.

The ordering on ${\mathbb{P}}_{2}$ is $\subseteq^{\ast}$.

It is shown in [VelOCA], see also [ShStvel], that ${\mathbb{P}}_{2}$ is
$\sigma$-directed closed. The following partial order was introduced in
[ShStvel] as a great tool to uncover the forcing preservation properties of
${\mathbb{P}}_{2}$, such as the fact that ${\mathbb{P}}_{2}$ is
$\aleph_{2}$-distributive (and so introduces no new $\omega_{1}$-sequences of
subsets of $\mathbb{N}$).

######  Definition 5.3 ([ShStvel]*2.2).

Given $\\{p_{\xi}:\xi\in\mu\\}$, define
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\mu\\})$ to be the partial order consisting
of all $q\in{\mathbb{P}}_{2}$ such that there is some $\xi\in\mu$ such that
$q=^{*}p_{\xi}$. The ordering on ${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\mu\\})$
is $p\leq q$ if $p\supseteq q$ (rather than $p\supseteq^{*}q$ for
${\mathbb{P}}_{2}$).

######  Lemma 5.4 ([14]).

In the forcing extension, $V[H]$, by $2^{<\omega_{1}}$, there is a maximal
$\subset^{*}$-descending sequence
$\\{p_{\xi}:\xi\in\omega_{1}\\}\subset{\mathbb{P}}_{2}$ which is
${\mathbb{P}}_{2}$-generic over $V$ and for which
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\omega_{1}\\})$ is ccc,
$\omega^{\omega}$-bounding, and preserves that $\mathbb{R}\cap V$ is not
meager.

######  Proposition 5.5.

If $\xi\in\omega_{1}$ and $\\{p_{\xi}:\xi\in\mu\\}$ is a descending sequence
in ${\mathbb{P}}_{2}$, then ${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\mu\\})$ is a
countable atomless poset.

######  Lemma 5.6 ([ShStvel]*2.4).

Given $\eta\in\omega_{1}$, a $\subset^{*}$-descending sequence
$\\{p_{\xi}:\xi\in\eta\\}\subset{\mathbb{P}}_{2}$, and a countable elementary
submodel $\mathfrak{U}\prec(H(\aleph_{2}),\in)$, such that
$\\{p_{\xi}:\xi\in\eta\\}\in\mathfrak{U}$, then there is a
$p\in{\mathbb{P}}_{2}$ which is $\mathfrak{U}$-generic for
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\eta\\})$. Moreover, for any extension
$\\{p_{\xi}:\xi\in\mu\\}\subset{\mathbb{P}}_{2}$ (again
$\subset^{*}$-descending) such that $\eta<\mu$ and $p_{\eta}=p$, every
$D\in\mathfrak{U}$ is predense in ${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\mu\\})$
provided it is dense in ${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\eta\\})$.

Almost all of the work we have to do is to establish additional preservation
results for the poset ${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\omega_{1}\\})$. The
sequence $\\{p_{\xi}:\xi\in\omega_{1}\\}$ (chosen in the forcing extension by
$2^{<\omega_{1}}$) will always be assumed to be ${\mathbb{P}}_{2}$-generic
over the PFA model. Following [ShStvel], let $\mathcal{F}$ be the filter on
${\mathbb{P}}_{2}$ generated by the sequence $\\{p_{\xi}:\xi\in\omega_{1}\\}$.
Note that $V[\mathcal{F}]$ is a generic extension of $V$ meaning that
$\\{p_{\xi}:\xi\in\omega_{1}\\}$ selects an element of every maximal antichain
of ${\mathbb{P}}_{2}$ that is an element of $V$ but it is chosen within the
model of CH and also introduces an $\omega_{1}$ sequence cofinal in
$\omega_{2}$. We may assume, as per [ShStvel], that many statements about
${\mathbb{P}}_{2}$-names of Borel subsets of $\mathbb{M}$ that are forced by
${\mathbb{P}}_{2}$ to hold will hold in $V[\mathcal{F}]$. In addition, there
are no new Borel subsets of $\mathbb{M}$.

Once these are established, we are able to apply the standard PFA type
methodology as demonstrated in [ShStvel, 14]. The technique is to construct a
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\omega_{1}\\})$-name $\dot{Q}$ of a proper
poset and to then invoke PFA in the ground model so as to select a filter
meeting a given choice of $\omega_{1}$-many dense subsets of
$2^{<\omega_{1}}*{\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\omega_{1}\\})*\dot{Q}$.
Using the proof that ${\mathbb{P}}_{2}$ is $\omega_{1}$-distributive, there is
then a condition $p\in{\mathbb{P}}_{2}$ that shows that simply forcing with
${\mathbb{P}}_{2}$ yields the desired conclusion from meeting those
$\omega_{1}$-many dense sets.

For example, it is shown in [14] that $\mathbb{P}$ forces that
$\mbox{Triv}(H)$ is a dense $P$-ideal (i.e. $(\dagger_{1}^{-})$ holds).
However, since this is weaker than $(\dagger_{1})$ we refer to the following
theorem to assert that $\mathbb{P}$ forces that $(\dagger_{1})$ holds.

######  Theorem 5.7 (4.17 of [Dowcccoverfin]).

In the extension obtained by forcing over a model of PFA, if $\Phi$ is a
homomorphism from $\mathcal{P}(\mathbb{N})/\mbox{fin}$ onto
$\mathcal{P}(\mathbb{N})/\mbox{fin}$, then $\mbox{Triv}(\Phi)$ is a ccc over
fin ideal.

Now we adapt the construction from [ShStvel] of
$\\{p_{\xi}:\xi\in\omega_{1}\\}$ so that we also have that $(\dagger_{2})$
holds. The approach also incorporates ideas from Velickovic’s proof that OCA
and $\mathbb{MA}(\omega_{1})$ implies $(\dagger_{1})$.

######  Theorem 5.8.

In the extension obtained by forcing over a model of PFA by
${\mathbb{P}}_{2}$, every $\mathbb{N}^{*}$ cut-set of $\mathbb{M}^{*}$ is
trivial on an ideal that is ccc over fin.

###### Proof.

Let $\\{\dot{\mathcal{C}}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$ be a family of
${\mathbb{P}}_{2}$-names that is forced (by $\mathbf{1}$) to be an
$\mathbb{N}^{*}$ cut-set of $\mathbb{M}^{*}$. Much as in Theorem 3.2, we can
assume that the family
$\\{\dot{\mathcal{C}}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$ is also forced to be
order-preserving in the sense that
$\left(\bigcup\dot{\mathcal{C}}_{f}\right)^{*}$ contains
$\left(\bigcup\dot{\mathcal{C}}_{g}\right)^{*}$ whenever $f<^{*}g$ are in
$\mathbb{N}^{\mathbb{N}}$.

Suppose also that the ideal of sets on which this cut-set is trivial is not
ccc over fin. Since ${\mathbb{P}}_{2}$ is $\aleph_{1}$-distributive, we may
choose a condition $p_{0}\in{\mathbb{P}}_{2}$ and an almost disjoint family
$\\{I_{\alpha}:\alpha\in\omega_{1}\\}$ of subsets of $\mathbb{N}$ such that
$p_{0}$ forces that none of the $I_{\alpha}$’s are in the trivial ideal for
this cut-set. By [VelOCA] (since this final model is a model of Martin’s
Axiom) it suffices to assume that the family
$\\{I_{\alpha}:\alpha<\omega_{1}\\}$ is tree-like. More specifically, there is
a function $\sigma:\mathbb{N}\rightarrow 2^{<\omega}$ such there is a 1-to-1
enumerated collection $\\{\rho_{\alpha}:\alpha<\omega_{1}\\}\subset
2^{\omega}$ so that for all $\alpha<\omega_{1}$,
$\sigma(I_{\alpha})\subset\\{\rho_{\alpha}\restriction j:j\in\omega\\}$.

To start the proof, let $p_{0}\in G$ be a ${\mathbb{P}}_{2}$-generic filter.
We consider the family $F$ of all partial functions $f\restriction I_{\alpha}$
($f\in\mathbb{N}^{\mathbb{N}}$ and $\alpha\in\omega_{1}$). We define the
relation $R\subset[F]^{2}$ to consist of all unordered pairs
$\\{f,f^{\prime}\\}$ that satisfying

  1. (1)

if $\operatorname{dom}(f)=I_{\alpha}$ and
$\operatorname{dom}(f^{\prime})=I_{\beta}$, then $\alpha\neq\beta$,

  2. (2)

there is an $m\in\operatorname{dom}(f)\cap\operatorname{dom}(f^{\prime})$ such
that $[c^{m}_{f},d^{m}_{f}]\cap[c^{m}_{f^{\prime}},d^{m}_{f^{\prime}}]$ is
empty.

We utilize the discrete topology on $S=\\{\infty\\}\cup\mathbb{Q}^{2}$ and for
any $f\in F$ we identify $f$ with an element, $s_{f}$, of the product space
$S^{\mathbb{N}}$, where in coordinate $m\in\operatorname{dom}(f)$,
$s_{f}(m)=(c^{m}_{f},d^{m}_{f})$ and for
$m\in\mathbb{N}\setminus\operatorname{dom}(f)$, $s_{f}(m)=\infty$. With this
topology, using tree-like as in [VelOCA], the relation $R$ is an open subset
of the product space $[\\{s_{f}:f\in F\\}]^{2}$ (i.e. unordered pairs). Assume
that $\Lambda$ is an uncountable subset of $\omega_{1}$ and that, for each
$\xi\in\Lambda$ we set $\bar{f}_{\xi}=f_{\xi}\restriction I_{\xi}$. The set
$\\{\bar{f}_{\xi}:\xi\in\Lambda\\}$ would be called an $R$-homogeneous set if
it satisfied that $\\{\bar{f}_{\xi},\bar{f}_{\eta}\\}\in R$ for all
$\xi\neq\eta\in\Lambda$. We show that $\\{\bar{f}_{\xi}:\xi\in\Lambda\\}$
fails to be an $R$-homogeneous set.

Choose any $f\in\mathbb{N}^{\mathbb{N}}$ so that $f_{\xi}<^{*}f$ for all
$\xi\in\Lambda$. Choose $\bar{m}\in\mathbb{N}$ and uncountable
$\Lambda_{1}\subset\Lambda$ so that
$[c^{m}_{f},d^{m}_{f}]\subset[c^{m}_{f_{\xi}},d^{m}_{f_{\xi}}]$ for all
$\bar{m}<m\in\mathbb{N}$. Choose also uncountable
$\Lambda_{2}\subset\Lambda_{1}$ so that
$\\{[c^{m}_{f_{\xi}},d^{m}_{f_{\xi}}]:m\leq\bar{m}\\}=\\{[c^{m}_{f_{\eta}},d^{m}_{f_{\eta}}]:m\leq\bar{m}\\}$
for all $\xi,\eta\in\Lambda_{2}$. These reductions ensure that
$\\{\bar{f}_{\xi},\bar{f}_{\eta}\\}$ is not in $R$ for any pair
$\xi,\eta\in\Lambda_{2}$.

Suppose now we prove that there is a sequence $\\{p_{\xi}:\xi\in\omega_{1}\\}$
as in Lemma 5.4 that, in addition, ensures that
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\omega_{1}\\})$ forces that there is a
proper poset $Q$ that does force there is an uncountable $R$-homogeneous set.
Then by our discussion in the paragraph following Lemma 5.6 we would have
contradicted that $\\{\dot{\mathcal{C}}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$ is
forced by ${\mathbb{P}}_{2}$ to be a cut-set.

Todorcevic [TodOCA] has shown that, since there is no uncountable
$R$-homogeneous set, the family $F\subset\mathbb{N}^{\mathbb{N}}$ must be
covered by a countable family $\\{F_{n}:n\in\omega\\}$ of sets each with the
property that $\left[F_{n}\right]^{2}$ is disjoint from $R$. Using our
separable metrizable topology on $F$ and the fact that $R$ is open, this is
equivalent to there being a family $\\{F_{n}:n\in\omega\\}$ of countable
subsets of $F$ satisfying that for every $n$, the set of pairs from the
closure of $F_{n}$ and that the union of the closures of the $F_{n}$’s cover
$F$. If we prove that $\\{p_{\xi}:\xi\in\omega_{1}\\}$ forces there is no such
sequence $\\{F_{n}:n\in\omega\\}$ then we have completed the proof of the
theorem. We use Lemma 5.6 to do so.

Let $\eta\in\omega_{1}$, $\\{p_{\xi}:\xi\in\eta\\}\subset{\mathbb{P}}_{2}$,
and the countable elementary submodel $\mathfrak{U}$ be as in Lemma 5.6. We
can assume further $\mathfrak{U}$ is equal to $M\cap H(\aleph_{2})$ for some
countable elementary submodel $M\prec H(\aleph_{3})$ satisfying that each of
the objects $\\{p_{\xi}:\xi\in\eta\\}$,
$\\{I_{\alpha}:\alpha\in\omega_{1}\\}$,
$\\{\dot{\mathcal{C}}_{f}:f\in\mathbb{N}^{\mathbb{N}}\\}$ are elements of $M$.
Let us note that $R$ itself is an element of $V$ and that we may assume there
is some $\xi<\eta$ such that $p_{\xi}$ forces that $R$ is the open set
resulting from the conditions (1) and (2) above applied to the family
$[F]^{2}$. It suffices to prove that if
$\\{\dot{F}_{n}:n\in\omega\\}\in\mathfrak{U}$ is a set of
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\eta\\})$-names of countable subsets of
$F$, then there is a choice of $p_{\eta}$ (as in Lemma 5.6) that forces this
sequence fails to have the covering properties mentioned in the previous
paragraph.

To do so, we first consider the forcing extension by the countable poset
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\eta\\})$. Since the $\dot{F}_{n}$’s are
countable names of subsets of $F$ we can fix a
$\delta\in\mathfrak{U}\cap\omega_{1}$ such that it is forced that every
element of $\bigcup\\{\dot{F}_{n}:n\in\omega\\}$ has as its domain an element
of $\\{I_{\beta}:\beta<\delta\\}$. There is a maximal antichain $E_{0}$ of
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\eta\\})$ (in $\mathfrak{U}$) where each
$p\in E_{0}$ forces one of the following statements

  1. (1)

there is an $f\in F\cap\mathfrak{U}$ such that $f$ is not in the closure of
$\dot{F}_{n}$ for each $n\in\omega$,

  2. (2)

there is an $n\in\omega$ and a pair $\\{f,f^{\prime}\\}\in\dot{F}_{n}\cap R$,

  3. (3)

each of the previous two statements (1) and (2) fail to hold.

Now consider any condition $q\in{\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\eta\\})$
that is an extension of an element of $E_{0}$ in which condition (3) holds.

Working in $M$ we can ask if, for each $f\in\mathbb{N}^{\mathbb{N}}$, there is
a condition $p_{f}<q$ in ${\mathbb{P}}_{2}(\\{p_{\xi}:\xi\in\eta\\})$ and an
integer $n_{f}$ satisfying that $f\restriction I_{\delta}$ is in the closure
of $\dot{F}_{n_{f}}$. We show that the answer to this question is “no”. If
such a pair, $p_{f},n_{f}$, exists for all $f\in\mathbb{N}^{\mathbb{N}}$, then
there would be a single condition $\bar{p}$ and integer $\bar{n}$ satisfying
that $\tilde{F}=\\{f\in F:p_{f}=\bar{p},n_{f}=\bar{n}\\}$ is $<^{*}$-cofinal
in $\mathbb{N}^{\mathbb{N}}$. Since $\bar{p}$ forces that
$\tilde{F}\restriction I_{\delta}$ is a subset of the closure of
$\dot{F}_{\bar{n}}$, it follows from the failure of (2) that
$[\tilde{F}\restriction I_{\delta}]^{2}$ is disjoint from $R$. But this means
that, for all $m\in I_{\delta}\setminus\bar{m}$, the family
$\\{[c^{m}_{f},d^{m}_{f}]:f\in\tilde{F}\\}$ is linked. Since a linked family
of compact intervals has non-empty intersection, this, together with the
cofinality of $F$, would imply that $I_{\delta}$ is in the trivial ideal.

So we have now proven there is some $f\in\mathbb{N}^{\mathbb{N}}$ such that
there is no pair $p_{f},n_{f}$ as above. By elementarity there is such an
$f\in M\cap{\mathbb{N}}^{\mathbb{N}}\subset\mathfrak{U}$. The failure of there
being a pair $p_{f},n_{f}$ is equivalent to the assertion, that for each
$n\in\omega$, there is a subset $E(q,f,n)$ of
${\mathbb{P}}_{2}(\\{p_{\xi}:\xi<\eta\\})$ that is dense below $q$ and for
each $e\in E(q,f,n)$, there is a basic open neighborhood of $f\restriction
I_{\delta}$ in the topology on $S^{\mathbb{N}}$ such that $e$ forces that
$\dot{F}_{n}$ is disjoint from this open set.

It follows then the choice of $p_{\eta}$ as in Lemma 5.6 will result in the
family $\\{\dot{F}_{n}:n\in\omega\\}$ considered as
${\mathbb{P}}_{2}(\mathcal{F})$-names will not be the dense sets for a
countable cover of $F$ consisting of sets whose set of pairs are disjoint from
$R$. The proof is completed, i.e. a contradiction reached, by applying the PFA
methodology discussed following Lemma 5.4. ∎

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