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\begin{document}

\title{Homeomorphism groups of manifolds\linebreak[1] and Erd\H{o}s space}

\author{Jan J. Dijkstra}
\address{Faculteit der Exacte Wetenschappen\,/\,Afdeling Wiskunde, Vrije Universiteit,
De Boelelaan 1081, 1081 HV\ \ Amsterdam, The Netherlands}
\email{dijkstra@cs.vu.nl}

\author{Jan van Mill}
\address{Faculteit der Exacte Wetenschappen\,/\,Afdeling Wiskunde, Vrije Universiteit,
De Boelelaan 1081, 1081 HV\ \ Amsterdam, The Netherlands}
\email{vanmill@cs.vu.nl}

\subjclass[2000]{Primary 57S05}

\date{September 30, 2003}

\commby{Krystyna Kuperberg}

\begin{abstract}Let $M$ be either a topological manifold, a Hilbert 
cube manifold,
or a Menger manifold and let $D$ be an arbitrary countable dense
subset of $M$. Consider the topological group $\mathcal{H}(M,D)$ which
consists of all autohomeomorphisms of $M$ that map $D$ onto itself
equipped with the compact-open topology. We present a complete
solution to the topological classification problem for $\mathcal{H}(M,D)$
as follows. If $M$ is a one-dimensional topological manifold, then
$\mathcal{H}(M,D)$ is homeomorphic to $\mathbb{Q}^\infty$, the countable power
of the space of rational numbers. In all other cases we found that
$\mathcal{H}(M,D)$ is homeomorphic to the famed Erd\H os space $\mathfrak
E$, which consists of the vectors in Hilbert space $\ell^2$ with
rational coordinates. We obtain the second result by developing
topological characterizations of Erd\H os space.
\end{abstract}

\maketitle

\section{Introduction}\label{introduction}
\emph{All spaces under discussion are separable and metrizable.} If $X$ is 
compact, then the standard
topology on the group of homeomorphisms $\mathcal{H}(X)$ of $X$ is the so-called compact-open
topology (which coincides with the topology of uniform convergence). This topology makes
$\mathcal{H}(X)$ a Polish topological group. For locally compact spaces, the compact-open topology
is Polish but not necessarily a group topology. We therefore think of $X$ as a subspace of its
Alexandroff one-point compactification $\alpha X = X\cup\{\infty\}$, and we topologize
$\mathcal{H}(X)$ by identifying it with the closed subgroup
    $\{h\in \mathcal{H}(\alpha X): h(\infty)=\infty\}$
of $\mathcal{H}(\alpha X)$. If every point in $X$ has a neighbourhood that 
is a continuum, 
then the
just described topology on $\mathcal{H}(X)$ coincides with the compact-open 
topology; see
Dijkstra~\cite{Di03} and Arens~\cite{arens46}. If $A$ is a subset of a 
space $X$, then
$\mathcal{H}(X,A)$ stands for the subgroup $\{h\in\mathcal{H}(X):h(A)=A\}$ of $\mathcal{H}(X)$.


Brouwer~\cite{Brouwer13a} showed that $\reals$ is countable dense 
homogeneous,  that is, for all
countable dense subsets $A$ and $B$ of $\reals$ there is an $h\in\MCH(\reals)$ with $h(A)=B$. It is
not difficult to prove that every $\reals^n$ has this property. %
In view of Brouwer's result it is a natural idea to investigate the group
$\mathcal{H}(\reals^n,\bbQ^n)$. It was shown in Dijkstra and van
Mill~\cite{DijkstravanMill03a} that the group $\mathcal{H}(\reals,\rationals)$ is homeomorphic to
the zero-dimensional space $\rationals^\infty$, the countable infinite product of copies of the
rational numbers $\rationals$. In contrast, we showed in \cite{DijkstravanMill03a} (see also
~\cite{Dijkstra03}) that $\mathcal{H}(\reals^n,\rationals^n)$ for $n \ge 2$ contains a closed copy
of the famed Erd\H{o}s space $\erdos$  which is known to be one-dimensional;  
see \cite{erdos}. This
result led us to consider the question whether $\mathcal{H}(\reals^n,\rationals^n)$ (for $n \ge 2$)
is in fact homeomorphic to Erd\H{o}s space.
We announce here that it is. We prove that if $D$ is a
countable dense subset of  a locally compact space $X$, then 
$\mathcal{H}(X,D)$ is an Erd\H{o}s
space factor, which means that $\mathcal{H}(X,D)\times \erdos$ is homeomorphic to $\erdos$. Under
rather mild extra conditions, the group $\mathcal{H}(X,D)$ is found to be 
homeomorphic to Erd\H{o}s
space. This is the case  if $X$ contains a nonempty open subset homeomorphic to $\reals^n$ for $n\ge
2$, an open subset of the Hilbert cube $Q$, or an open subset of some universal Menger continuum. As
an application it follows that if $M$ is an at least 2-dimensional manifold (with or without
boundary) and $D$ is a countable dense subset of $M$, then $\mathcal{H}(M,D)$ 
is homeomorphic to
Erd\H{o}s space.

Homeomorphism groups of manifolds are  very well studied. Let $\interval$ denote the interval
$[0,1]$ and let $\MCH_{\Bd}(\bbI^n)$ stand for the subgroup of $\MCH(\bbI^n)$ consisting of homeomorphisms
that fix the boundary of the $n$-cube $\bbI^n$. Anderson~\cite{and:homeomorphismgroup} proved that
$\mathcal{H}_\Bd(\interval)$ is homeomorphic to the separable Hilbert space $\ell^2$ (see
\cite[Proposition VI.8.1]{bepe75} or \cite{keesling71}). It was shown by {Luke} and
{Mason}~\cite{LukeMason72} that $\mathcal{H}_\partial(\interval^2)$ is an absolute retract, which
implies that $\mathcal{H}_\partial(\interval^2)\approx\ell^2$ (apply for instance {Dobrowolski} and
{Toru\'nczyk}~\cite{doto81}). For $n\ge 3$ it is open whether $\mathcal{H}_\partial(\interval^n)$ is
an absolute retract. This is one of the most interesting open problems in infinite-dimensional
topology. For the Hilbert cube $Q$, that is, for $n=\infty$, the analogous problem was solved by
{Ferry}~\cite{ferry77} and  {Toru\'nczyk}~\cite{Torunczyk77}. They proved that $\mathcal{H}(Q)$ is
homeomorphic to $\ell^2$ (observe that $Q$ has no boundary). For $3\le n < \infty$ it is unknown
what the topological classification of  $\mathcal{H}_\Bd(\interval^n)$ or $\mathcal{H}(\interval^n)$
is. By our results, the subgroups $\mathcal{H}_\Bd(\interval^n,(\rationals\cap\interval)^n)$ and
$\mathcal{H}(\interval^n,(\rationals\cap\interval)^n)$ \emph{are} known; 
they are homeomorphic to
Erd\H{o}s space.

Recall that the \emph{Erd\H{o}s space \erdos} is the `rational Hilbert space', that is the set of
vectors in $\ell^2$  the coordinates of which are all rational. This space was introduced by Hurewicz who
asked to compute its dimension. Erd\H{o}s \cite{erdos} proved that $\erdos$ is one-dimensional by
establishing that every nonempty clopen subset of \erdos\ is unbounded. This result, in combination
with the obvious fact that $\erdos$  is homeomorphic to $\erdos\times\erdos$, lends the space its
importance in dimension theory. \emph{Complete Erd\H{o}s space \cerdos} is the `irrational Hilbert
space', that is, the set of vectors in $\ell^2$ all coordinates of which 
are irrational. In contrast
to \erdos, the complete Erd\H{o}s space is topologically complete, being a $G_\delta$-subset of
$\ell^2$. The space \cerdos\ surfaced in topological dynamics as the `endpoint' set of several
interesting objects. See  Kawamura, Oversteegen, and Tymchatyn~\cite{KaOvTy96} for more information.

In order to prove our results  we first present several
increasingly powerful topological characterizations of Erd\H{o}s
space. What sets Erd\H os space apart from familiar spaces is that
in addition to the one-dimensional topology that it inherits from
$\ell^2$, an important role is played by the zero-dimensional
topology that $\erdos$ inherits from the product space
$\bbQ^\infty$. This bitopological aspect prompted us to define and
develop several new concepts in topology that link the two
topologies in the characterization theorems. We  demonstrate the
power of our characterizations by deriving from them the above
results with relative ease. Along the way, we get several other
interesting results. For example, Erd\H{o}s space is homeomorphic
to its countable infinite power. Here we have a striking contrast
with $\cerdos$, which is \emph{not} homeomorphic to
$\cerdos^\infty$, as was proved by Dijkstra, van Mill, and
Stepr\=ans~\cite{DivMSt}. In addition, Erd\H{o}s space is
homeomorphic to $\cerdos\times\bbQ^\infty$, and every nonempty open
subset of $\erdos$ is homeomorphic to $\erdos$.

We conclude with the observation that Erd\H{o}s space started its career as a curious example in
dimension theory. It turns out however that it is a fundamental object that surfaces in many places.
In addition, it allows for a useful and easily 
applied topological characterization just as several other
fundamental objects in topology: the Cantor set (Brouwer~\cite{br:perfect}), the Hilbert cube
(Toru\'nczyk~\cite{tor:hilbertcube}),  Hilbert space (Toru\'nczyk~\cite{tor:hilbertspace}), the
universal Menger continua (Bestvina~\cite{bestvina}), and the N\"obeling spaces
(Ageev~\cite{Ageev02}).

Detailed proofs supporting the theorems we announce here will appear in \cite{DijkstravanMill03a}
 for \S2, \cite{DijkstravanMill03b} for \S\S3--5, and \cite{Dijkstra03} for \S5.

\section{The zero-dimensional case}
In this section we consider $\MCH(M,D)$, where $M$ is a one-dimensional topological manifold.
 We
recall the following characterization of the space $\bbQ^\infty$,
which follows from a theorem of Steel~\cite{st80}; see also
 van Engelen~\cite[Theorem A.2.5]{fons86}.

\begin{theorem}\label{QNchar} A
 space $X$ is homeomorphic to $\bbQ^\infty$ if and only if
$X$ is a zero-dimensional,  first category $F_{\sigma\delta}$-space
 with the property that no nonempty clopen subset is a
$G_{\delta\sigma}$-space.
 \end{theorem}
In particular, we have:
 \begin{corollary}\label{QNcorr}If $X$ is a homogeneous,
zero-dimensional, first
 category $F_{\sigma\delta}$-space that contains a closed copy of\/
 $\bbQ^\infty$, then $X$ is homeomorphic to $\bbQ^\infty$.
 \end {corollary}

The following theorem is proved by constructing a closed imbedding
of $\bbQ^\infty$ in $\MCH(\bbR,\bbQ)$.
\begin{theorem}\label{RQ}  $\MCH(\bbR,\bbQ)$ is homeomorphic to
$\bbQ^\infty$.
\end{theorem}

\begin{corollary}\label{RQgen} Let $D$  be a countable dense subset of  a locally compact space $X$.
    If $X$ contains an open set that is  homeomorphic to $\bbR$, then $\MCH(X,D)$ is homeomorphic to $\bbQ^\infty$
 if and only if $\MCH(X,D)$ is zero-dimensional.
\end{corollary}
We also showed that if $D$ is a countable dense subset of a Cantor
set $C$, then $\MCH(C,D)\approx\bbQ^\infty$.

\section{Almost zero-dimensional spaces} Let $p\in(0,\infty)$ and consider the (quasi-)Banach
space $\ell^p$. This space consists of all sequences $z=(z_0,z_1,z_2,\dotsc)\in\bbR^\infty$ such that
$\sum_{i=0}^\infty|z_i|^p<\infty$.  The topology on $\ell^p$ is generated by the norm
$\|z\|=(\sum_{i=0}^\infty|z_i|^p)^{1/p}$.  It is well known that  the norm topology on $\ell^p$ is
generated by the product topology (that is inherited from $\bbR^{\infty}$) together with the sets
$\{z\in\ell^2\colon \|z\|0$. We extend the $p$-norm over $\bbR^\infty$ by putting
$\|z\|=\infty$ when $z\in\bbR^\infty\smin\ell^p$. Note also that the norm as a function from
$\bbR^\infty$ to $[0,\infty]$ is not continuous because the norm topology is much stronger than the
product topology, but that this function is lower semi-continuous (LSC).
 We define the {\em Erd\H os space\/} by
    \[\erdos=\{z\in\ell^2\colon\text{$z_i\in\bbQ$ for each
$i$}\}.\]
    Let $\mathcal T$ stand for the zero-dimensional topology that $\erdos$ inherits from
$\bbQ^\infty$. Observe that $\mathcal T$ is weaker than the norm
topology, and hence that $\erdos$ is totally disconnected.
We have by the remark above that the graph of the norm function, when seen
as a function from $(\erdos,\mathcal T)$ to $\bbR^+=[0,\infty)$, is
homeomorphic to $\erdos$. So, informally, we can think of \erdos\
as a `zero-dimensional space with some LSC function declared
continuous'. We find it convenient to work with USC rather than
LSC functions, and we therefore define $\eta:\bbQ^\infty\to\bbR^+$ by
$\eta(z)=1/(1+\|z\|)$, where $1/\infty=0$.

There is an interesting connection between the two topologies on
\erdos\ that we would like to draw attention to. Because the norm
is LSC on $\bbR^\infty$, every  closed $\eps$-ball in $\erdos$ is
also closed in the zero-dimensional space $\bbQ^\infty$.  Thus we have that
every point in $\erdos$ has arbitrarily small
neighbourhoods which are intersections of clopen sets.

A subset $A$ of a space $X$ is called a {\em C-set in\/} $X$ if $A$ can be written as an
intersection of clopen subsets of $X$. A space  is called {\em almost zero-dimensional\/} if every
point of the space has a neighbourhood basis consisting of C-sets of the space. This concept is due
to Oversteegen and Tymchatyn~\cite{OvTy94}. The definition we use here is different from the
original one but its equivalence is established in Dijkstra, van Mill, and Stepr\=ans~\cite{DivMSt}.
Note that almost zero-dimensionality is hereditary. It is proved in \cite{OvTy94} that every almost
zero-dimensional space is at most one-dimensional; see also Levin and Pol~\cite{LevinPol}.

Thus  $\erdos$ is almost zero-dimensional. In fact, it is a universal object for the class of almost
zero-dimensional spaces (in contrast, the class of totally disconnected spaces has no universal
element; see Pol~\cite{Pol73}):

\begin{theorem}\label{AZDchar}
The following statements about a space $X$ are equivalent:
\begin{enumerate}
    \item $X$ is almost zero-dimensional,
    \item $X$ is homeomorphic to
the graph of some USC or LSC function with  a domain of dimension at most zero,
    \item $X$ is imbeddable in complete Erd\H os space $\cerdos$, and
    \item $X$ is imbeddable in Erd\H os space $\erdos$.
\end{enumerate}
\end{theorem}
This theorem can be extracted from results in the papers \cite{OvTy94} and \cite{KaOvTy96}.
 Particularly important is the characterization theorem in \cite{OvTy94}
 that states that a space is almost zero-dimensional if and only if it is
 homeomorphic to
the set of endpoints of some $\reals$-tree.
 As a corollary to Theorem~\ref{AZDchar} we have that the nonempty
C-sets in an almost zero-dimensional space are precisely the retracts of the space.

If $Z$ is a set that contains $X$, then we say that a (separable
metric) topology $\mathcal{T}$ on $Z$ \emph{witnesses the almost
zero-dimensionality of} $X$ if $\dim(Z,\mathcal{T})\le 0$, $O\cap
X$ is open in $X$ for each $O\in \mathcal{T}$, and every point of
$X$ has a neighbourhood basis in $X$ consisting of sets that are
closed in $(Z,\mathcal{T})$. We will also say that the space
$(Z,\mathcal{T})$ is a witness to the almost zero-dimensionality
of $X$. The archetype is $\bbQ^\infty$ as a witness to the almost
zero-dimensionality of Erd\H os space.

\section{Characterizing Erd\H{o}s space topology}
Let $\varphi,\psi\colon X\to \reals^+$ be such that $\psi(x)\le
\varphi(x)$ for all $x\in X$. We define
    \[G^\varphi_\psi=\{(x,\varphi(x))\colon x\in X\hbox{
and } \varphi(x)>\psi(x)\}
    \]
and
    \[L^\varphi_\psi=\{(x,t)\colon x\in X\hbox{ and }\psi(x)\le t\le
\varphi(x)\},\]
     both equipped with the topology inherited from
$X\times\reals^+$. Observe that $G^\eta_0$ is homeomorphic to
$\erdos$.

We say that $\varphi$ is a {\em Lelek function with bias\/} $\psi$ if $X$ is zero-dimensional,
$\varphi$ and $\psi$ are USC, $X'=\{x\in X\colon\psi(x)<\varphi(x)\}$ is dense in $X$, and
$G^\varphi_\psi$ is dense in $L^{\varphi\restriction X'}_{\psi\restriction X'}$. If $\varphi$ is a
Lelek function with bias $0$, then $\varphi$ is simply called a {\em Lelek function}. This
terminology finds its origin in the following fact. If $\vphi$ is a Lelek function with compact
domain $C$, then we obtain a Lelek fan~\cite{Le61} by identifying the base $C\times\{0\}$ in
$L^\vphi_0$ to a point.
 Observe that
$\eta$ is a Lelek function.

 If $A$ is a nonempty set, then $A^{<\omega}$ denotes the set of
all finite strings of elements of $A$, including the null string $\lambda$.  Let $A^\omega$ denote
the set of all infinite strings of elements of $A$. If $s\in A^{<\omega}$ and $\sigma\in
A^{<\omega}\cup A^\omega$, then we put $s\prec\sigma$ if $s$ is an initial substring of $\sigma$. If
$\sigma\in A^{<\omega}\cup A^\omega$ and $k\in\omega$, then $\sigma\restr k\in A^{<\omega}$ is the
string of length $k$ with $\sigma\restr k\prec \sigma$. A {\em tree} $T$ over a set $A$ is a subset
of $A^{<\omega}$ that is closed under initial segments, that is, if $s\in T$ 
and $t\prec s$, then
$t\in T$.  An {\em infinite branch} of $T$ is an element $\sigma$ of $A^\omega$ such that
$\sigma\restr k\in T$ for every $k\in \omega$. The {\em body} of $T$, 
written as $[T]$, is the set of
all infinite branches of $T$. If $s\in T$, then $\succc(s)$ denotes the set of immediate successors of
$s$ in $T$.

Sierpi\'nski~\cite{sierpinski24} has shown that  $X$ is an (absolute) $F_{\sigma\delta}$-space if
and only if there exists a nonempty tree $T$ over a countable set and closed subsets $X_s$ of $X$
for each $s\in T$ such that:
    \begin{enumerate}
    \item[i.]\label{sierpeen} $X_\lambda=X$ and  $ X_s=\bigcup\{X_t\colon
t\in\succc(s)\}$ for all $s\in T$ and
    \item[ii.]\label{sierptwee} if
$\sigma\in[T]$, then the sequence $
    X_{\sigma\restriction 0}, X_{\sigma\restriction 1},\dotsc$ converges to a point $x_\sigma\in
X$.
\end{enumerate} Let us call such a system $(X_s)_{s\in T}$ a {\em
Sierpi\'nski stratification\/} of $X$. Van Engelen~\cite[Theorem
A.1.6]{fons86} has shown that a zero-dimensional space $X$ is
homeomorphic to $\rationals^\infty$ if there exists a Sierpi\'nski
stratification $(X_s)_{s\in T}$ of $X$ such that $X_t$ is nowhere
dense in $X_s$ whenever $t\in\succc(s)$. Our characterizations of
\erdos\ were inspired by these results.

\begin{definition}\label{SLprime}
$\mathsf{SL}$ is the class of all bounded USC functions
$\vphi\from X\to \reals^+$ such that $X$ is a zero-dimensional
space for which there exists a Sierpi\'nski stratification
$(X_s)_{s\in T}$  with the following properties:
    \begin{enumerate}
    \item[(a)]  if $s\in T$ and
$t\in \succc(s)$, then $G^{\vphi\restr X_{t}}_0$ is nowhere
    dense in $G^{\vphi\restr X_s}_0$ and
    \item[(b)] if $s\in T$, then $\vphi\restr X_s$ is a Lelek
function.
    \end{enumerate}
\end{definition}
If we define $T=\bbQ^{<\omega}$ and $X_{q_1\dotsc
q_k}=\{q_1\}\times\dots\times\{q_k\}\times\bbQ\times\bbQ\times\dotsb$,
then it is a straightforward exercise to show that
$\eta:\bbQ^\infty\to\bbR^+$ is an element of $\mathsf{SL}$. We call
a pair $(h,\beta)$ a {\em homeomorphism} from a function
$\vphi:X\to\bbR^+$ to a function $\psi:Y\to\bbR^+$ if
 $h: X\to Y$ is a homeomorphism and $\beta:X\to(0,\infty)$ is a continuous map
  with $\psi\circ h=\beta\dt\vphi$.
\begin{theorem}\label{paar}
Any two elements of\/ $\mathsf{SL}$ are homeomorphic and hence a space $E$ is homeomorphic to
$\erdos$ if and only if $E\approx G^\vphi_0$ for some $\vphi\in\mathsf{SL}$.
\end{theorem}

We sketch the method by which we proved this theorem. Let
$\vphi:X\to\bbR^+$ and $\psi:Y\to\bbR^+$ be elements of $\mathsf{SL}$
and let $(X_t)_{t\in T}$ and $(Y_s)_{s\in S}$ be the associated
Sierpi\'nski stratifications. We construct a boolean algebra of
clopen subsets of $X$ such that the associated Stone space $C$ is
a compactification of $X$ that admits an extension
$\tilde\vphi:C\to\bbR^+$ of $\vphi$ with the property that
$\tilde\vphi\res \ovl X_t$ is a Lelek function for each $t\in T$.
The function $\psi$ is similarly extended to
$\tilde\psi:D\to\bbR^+$. Van Engelen's proof \cite[pp.\
115--120]{fons86} of the characterization of $\bbQ^\infty$ in
terms of Sierpi\'nski stratifications shows that there exists a
homeomorphism $h:C\to D$ with $h(X)=Y$. This homeomorphism in
general will not correspond to a homeomorphism between the functions
$\tilde\vphi$ and $\tilde\psi$. In order to get a continuous
$\beta:C\to(0,\infty)$ such that $\tilde\psi\circ
h=\beta\dt\tilde\vphi$ and $h(X)=Y$ we need to add an additional
ingredient to van Engelen's construction as follows. The
uniqueness of the Lelek fan as proved by Bula and
Oversteegen~\cite{BulaOversteegen} and
Charatonik~\cite{Charatonik89} allows us to develop an
`unknotting' theory for Lelek functions which does the trick:
\begin{theorem}[Homeomorphism Extension]\label{homeoext}
Let $\vphi:C\to\bbR^+$ and $\psi:D\to\bbR^+$ be Lelek functions with compact domain. Let $A\subset C$
and $B\subset D$ be closed sets such that $G^{\vphi\restr A}_0$ and $G^{\psi\restr B}_0$ are nowhere
dense in $G^\vphi_0$, respectively $G^{\psi}_0$. Then every homeomorphism from $\vphi\res A$ to
$\psi\res B$ can be extended to a homeomorphism from $\vphi$ to $\psi$ $($with control\/$)$.
\end{theorem}

Although it is a rather elegant characterization, Theorem~\ref{paar} is
not powerful enough because it is an `external' or `positional'
characterization in two ways. First, in order to apply the theorem
to an Erd\H os space candidate $E$ the space has to come equipped
with a particular Lelek function $\vphi$ such that $E\approx
G^\vphi_0$, that is, $E$ has to come positioned in the space
$\bbQ^\infty\times\bbR^+$. In addition, any function
$\vphi\in\mathsf{SL}$ assumes the value 0 (look for instance at
$\eta$), which means that $E$ will always correspond to a {\em
proper} subset of the graph of $\vphi$.  Our next characterization
will be `internal'. In order to formulate the internal properties
of a space that will guarantee the existence of Lelek functions
such as $\vphi$ we need to introduce some new concepts.

As was mentioned in \S1, Erd\H os~\cite{erdos} proved that every nonempty clopen subset of \erdos\
is unbounded. This means that every vector in \erdos\ has a neighbourhood that does not contain any
nonempty clopen subsets of $\erdos$. This property of \erdos\ turns out to be crucial, and we
formalize it as follows.

\begin{definition}Let $X$ be a space and let $\mathcal{A}$ be a collection of subsets of $X$.
The space $X$ is called \emph{$\mathcal{A}$-cohesive\/} if every point of the space has a
neighbourhood that does not contain nonempty clopen subsets of any element of $\mathcal{A}$. If a
space $X$ is $\{X\}$-cohesive, then we simply call $X$ \emph{cohesive}.
\end{definition}


A cohesive space is obviously at least one-dimensional at every point, but it is easily seen that the
converse is not valid. However, the situation is simple for topological groups because a topological
group is cohesive if and only if it is not zero-dimensional.





\begin{definition}\label{anchor} Let $T$ be a tree and let
$(X_s)_{s\in T}$ be a system of subsets of a space $X$ such that $X_t\subset X_s$ whenever $s\prec
t$. A subset $A$ of $X$ is called an {\em anchor} for $(X_s)_{s\in T}$ in $X$ if for every
$\sigma\in[T]$ we have either $X_{\sigma\restriction k}\cap A=\emptyset$ for some $k\in \omega$ or
the sequence $X_{\sigma\restriction 0},X_{\sigma\restriction 1},\dotsc$ converges to a point in $
X$.
\end{definition}
Thus the anchor $A$ has the property that for every sequence that is generated by an element of
$[T]$, if it is attached to $A$, then it must converge and cannot be free to drift out of the space.
Note that if $(X_s)_{s\in T}$ is a Sierpi\'nski stratification, then the whole space is an anchor.


We now present our first internal characterization of $\erdos$.

\begin{definition}\label{Ee} $\mathsf E$ is the class of all nonempty spaces
$E$ such that there exists a topology $\mathcal{T}$ on $E$ that
witnesses the almost zero-dimensionality of $E$ and there exist a
nonempty tree $T$ over a countable set and subspaces $E_s$ of $E$
that are closed with respect to $\mathcal{T}$ for each $s\in T$
such that:
\begin{enumerate}\label{voorlaatstedef}
    \item\label{Eeen} $E_\lambda=E$ and $E_{s}=\bigcup\{E_{t}\colon t\in\succc(s)\}$ whenever  $s\in T$,
   \item\label{Etwee}  each $x\in E$ has a neighbourhood $U$ that is
        an anchor for $(E_s)_{s\in T}$ in $(E,\mathcal{T})$,
    \item\label{Edrie}  for each $s\in T$
and $t\in \succc(s)$ we have that $E_t$ is nowhere
    dense in $E_s$, and
    \item\label{Evier} $E$ is $\{E_s\colon s\in T\}$-cohesive.
\end{enumerate}
\end{definition}


\begin{theorem}\label{single}
$\mathsf E$ is the class of all spaces that are homeomorphic to $\erdos$.
\end{theorem}
We sketch the essence of the proof. Let $E\in\mathsf E$ and let $\mathcal T$ and $(E_s)_{s\in T}$ be
the associated witness topology and stratification. It suffices to show that there is a
$\chi\in\mathsf{SL}$ with $E\approx G^\chi_0$. Let $Z$ denote the zero-dimensional space
$(E,\mathcal T)$. We begin by taking the stratification $(E_s)_{s\in T}$ through a `refining'
process such that $Z$ admits a zero-dimensional extension $X$ with the property that $(\ovl
E_s)_{s\in T}$ becomes a Sierpi\'nski stratification of $X$, that is, the whole space becomes an
anchor. Consider now the space $Y$ that corresponds to the set $X$ with the topology that is
generated by the union of the topologies of $X$ and $E$. Then $X$ is a witness to the almost
zero-dimensionality of $Y$ and $E$ is an open subspace of $Y$. With Theorem~\ref{AZDchar} we can find a USC
function $\vphi$ the graph of which is homeomorphic to $Y$. By careful construction we can arrange that
$\vphi:X\to\bbI$ and that $G^\vphi_0\approx E$. Condition (4) of Definition~\ref{voorlaatstedef} allows us to
construct a USC function $\psi:X\to\bbI$ such that $G^\vphi_0=G^\vphi_\psi$ and $\vphi\res \ovl E_s$
is a Lelek function with bias $\psi\res\ovl E_s$ for each $s\in T$. We then remove the bias by
replacing the pair $(\vphi,\psi)$ by $(\chi,0)$ such that $G^\vphi_\psi\approx G^\chi_0$ and
$\chi\res\ovl E_s$ is a Lelek function for each $s\in T$. We now have that $\chi\in\mathsf {SL}$ and
can apply Theorem~\ref{paar}.



\begin{remark}The anchor concept in Definition~\ref{Ee} is essential. This is because condition (4)
excludes the possibility that the whole $E$ is an anchor for $(E_s)_{s\in T}$, 
that is,  we cannot
have a Sierpi\'nski stratification.

Note that in our characterization theorems we need a particular witness topology on the space. Of
course this topology is not uniquely determined. So one might ask why we do not use the `witness
topology' $\mathcal T$ that is generated by \emph{all\/} clopen subsets of $E$. The reason is that
this topology is not metrizable for Erd\H os space. Specifically, whenever $E$ is almost
zero-dimensional and cohesive, then $\mathcal T$ has uncountable character at every point.
\end{remark}
Our final and most powerful characterization of Erd\H os space is captured by the following
definition.

\begin{definition}\label{laatstedef}
$\mathsf E'$ is the class of all nonempty spaces $E$ such that
there exists an $F_{\sigma\delta}$-topology $\mathcal{T}$ on $E$
that witnesses the almost zero-dimensionality of $E$ and there
exist a nonempty tree $T$ over a countable set and subspaces $E_s$
of $E$ that are closed with respect to $\mathcal{T}$ for each
$s\in T\smin\{\lambda\}$ such that:
\begin{enumerate}
    \item[$(1')$]\label{Epeen} $E_\lambda$ is dense in $E$ and
    $E_{s}=\bigcup\{E_{t}\colon t\in\succc(s)\}$ whenever  $s\in T$,
   \item[$(2')$]\label{Eptwee}  each $x\in E$ has a neighbourhood $U$
that is an anchor for $(E_s)_{s\in T}$ in $(E,\mathcal{T})$,
    \item[$(3')$]\label{Epdrie}  for each $s\in T\smin\{\lambda\}$
and $t\in \succc(s)$ we have that $E_t$ is nowhere
    dense in $E_s$,
    \item[$(4')$]\label{Epvier} $E$ is $\{E_s\colon s\in
    T\}$-cohesive, and
    \item[$(5')$]\label{Epvijf} $E$ can be written as a countable union of
nowhere dense subsets that are closed with respect to
$\mathcal{T}$.
\end{enumerate}
\end{definition}

\begin{theorem}\label{EEprime}
$\mathsf E'=\mathsf E=\{E\colon E\approx\erdos\}$.
\end{theorem}
The inclusion $\mathsf E\subset\mathsf E'$ is a triviality. Let us 
consider an element $E$ of
$\mathsf E'$ with associated topology $\mathcal T$ and stratification $(E_t)_{t\in T}$. Since
$\mathcal T$ is an absolute $F_{\sigma\delta}$-topology we can find a Sierpi\'nski stratification
$(Z_s)_{s\in S}$ for $(E,\mathcal T)$. The proof now consists in carefully `grafting' the
stratification $(E_t)_{t\in T}$ onto $(Z_s)_{s\in S}$ so that the combined stratification satisfies
Definition~\ref{Ee}.


\begin{remark}\label{univ} At first glance there does not appear to be much difference between
Definitions~\ref{Ee}
and \ref{laatstedef}. This, however, is a false impression. To use Theorem~\ref{single} we have to
construct a stratification of the entire space, whereas condition $(1')$ of
Definition~\ref{laatstedef} requires only a stratification of a dense subset of $E$. Let us examine
the consequences  if $E$ is for instance a topological group. Then we need only three things to
satisfy Definition~\ref{laatstedef}: an $F_{\sigma\delta}$ witness topology, the first category
property $(5')$, and a suitable closed imbedding of Erd\H os space in $E$. 
Because if we have  a
copy $\mathcal E$ of $\erdos$ in $E$ of the right type, which means 
in particular that it is also a
closed imbedding on the level of the respective witness topologies, then we can obtain the dense
stratified set $E_\lambda$ by simply multiplying $\mathcal E$ with a countable dense subset of the
group $E$. This is the method that we will use to classify homeomorphism groups. In effect,
Theorem~\ref{EEprime} allows for a universality type argument similar to 
those used in zero-dimensional
(cf.\ Corollary \ref{QNcorr}) and infinite-dimensional topology.
\end{remark}
\section{Applications}
The following result follows easily from Theorem~\ref{single}.
\begin{proposition} If $A\subset\erdos$ is either nonempty and open or the complement of a
$\sigma$-compactum, then $A$ is homeomorphic to $\erdos$.
\end{proposition}

The following lemma can be found in Dijkstra~\cite{Dijkstra03} and is a straightforward
generalization of Erd\H os~\cite{erdos}. Recall that if $A_0,A_1,\dotsc$ is a sequence of subsets of
a space $X$, then $\limsup_{n\to\infty}A_n=\bigcap_{n=0}^\infty\CL{\bigcup_{k=n}^\infty A_k}$.
Consider the space $\ell^p$.
\begin{lemma}\label{erdos1} Let $E_0, E_1,E_2,\dotsc$ be a
sequence of subsets of $\reals$ such that  $0$ is a cluster point
of\/ $\limsup_{n\to\infty} E_n$.
 If we define
\begin{equation*}
    \mathcal{E}=\{z\in\ell^p\colon z_n\in E_n\text { for every
    $n$}\},
\end{equation*}
 then every nonempty clopen subset of $\mathcal{E}$ is
unbounded $($and hence $\mathcal{E}$ is cohesive$)$.
\end{lemma}

The following results show that there is great flexibility in the construction of $\erdos$.
\begin{proposition}\label{erdosSL}
Let $\mathcal{E}$ be a nonempty space as in Lemma~\ref{erdos1} such
that every $E_n$ is an $F_{\sigma\delta}$-space that is
zero-dimensional. If infinitely many of the $E_n$'s are of the
first category in themselves, then $\mathcal{E}\in\mathsf{E}$. Thus
$\mathcal{E}$ is homeomorphic to $\erdos$.
\end{proposition}
\begin{corollary}
$\cerdos\times \rationals^\infty$ is homeomorphic to $\erdos$. \end{corollary}
 A space $X$ is called
an {\em Erd\H os space factor\/} if there exists a space $Y$ with $X\times Y\approx\erdos$. The
following characterization follows easily from Theorem~\ref{single}.

\begin{theorem}\label{factor}
For a  nonempty space $E$ the following statements are equivalent:
\begin{enumerate}
    \item $E\times \erdos$ is homeomorphic to $\erdos$,
    \item $E$ is an
Erd\H os space factor,
    \item $E$ admits a closed imbedding into $\erdos$,
    \item $E$ is homeomorphic to a $G_\delta$-subset of $\erdos$,
and
    \item $E$ is almost zero-dimensional as
witnessed by an $F_{\sigma\delta}$-topology.
\end{enumerate}
\end{theorem}
\begin{corollary}
$\erdos^\infty$ is homeomorphic to $\erdos$. \end{corollary}


Our main applications are the ones on homeomorphism groups mentioned in \S\ref{introduction}.
 The first step in satisfying Definition~\ref{laatstedef} is to find an $F_{\sigma\delta}$ witness topology.
 If $D$
is a countable dense subset of a compact space $X$, then the topology of pointwise convergence on $D$
turns out to be precisely the right witness to the almost zero-dimensionality of $\mathcal{H}(X,D)$.
In view of Theorem~\ref{factor} we now have:

\begin{theorem}\label{Fsdgroup}
If  $D$ is  a countable dense subset of a locally compact space $X$, then $\mathcal{H}(X,D)$ is an
Erd\H os space factor.
\end{theorem}

We use  Theorem~\ref{EEprime} to prove our main application, as follows.

\begin{theorem}\label{hoofdstelling}
Let $D$ be a countable dense subset of a locally compact space $X$. If $X$ contains an open set that
is a topological $n$-manifold with
    $n\ge2$, a Hilbert cube manifold, or a manifold modelled on a
    universal Menger continuum,
 then $\mathcal{H}(X,D)$ is homeomorphic to Erd\H os space.
\end{theorem}

To use the method outlined in Remark~\ref{univ} we need suitable imbeddings 
of
Erd\H os space in
$\MCH(X,D)$. Fortunately, when we arrived at this point, it turned out that we already constructed
the right imbeddings in Dijkstra and van Mill~\cite{DijkstravanMill03a} and
Dijkstra~\cite{Dijkstra03} for the purpose of showing that the homeomorphism groups in question are
one-dimensional.


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\end{document}