Universal autohomeomorphisms of $\mathbb{N}^*$

Abstract

We study the existence of universal autohomeomorphisms of $\mathbb{N}^*$. We prove that the Continuum Hypothesis ($\mathsf{CH}$) implies there is such an autohomeomorphism and show that there are none in any model where all autohomeomorphisms of $\mathbb{N}^*$ are trivial.

Introduction

This paper is concerned with universal autohomeomorphisms on $\mathbb{N}^*$, the Čech-Stone remainder of $\mathbb{N}$.

In very general terms we say that an autohomeomorphism $h$ on a space $X$ is universal for a class of pairs $(Y,g)$, where $Y$ is a space and $g$ is an autohomeomorphism of $Y$, if for every such pair there is an embedding $e:Y\to X$ such that $h\circ e=e\circ g$, that is, $h$ extends the copy of $g$ on $e[Y]$.

In 1, Section 3.4 one finds a general way of finding universal autohomeomorphisms. If $X$ is homeomorphic to $X^\omega$ then the shift mapping $\sigma :X^\mathbb{Z}\to X^\mathbb{Z}$ defines a universal autohomeomorphism for the class of all pairs $(Y,g)$, where $Y$ is a subspace of $X$. One embeds $Y$ into $X^\mathbb{Z}$ by mapping each $y\in Y$ to the sequence $\langle g^n(y):n\in \mathbb{Z}\rangle$.

Thus, the Hilbert cube carries an autohomeomorphism that is universal for all autohomeomorphisms of separable metrizable spaces and the Cantor set carries one for all autohomeomorphisms of zero-dimensional separable metrizable spaces. Likewise the Tychonoff cube $[0,1]^\kappa$ carries an autohomeomorphism that is universal for all autohomeomorphisms of completely regular spaces of weight at most $\kappa$, and the Cantor cube $2^\kappa$ has a universal autohomeomorphism for all zero-dimensional such spaces.

Our goal is to have an autohomeomorphism $h$ on $\mathbb{N}^*$ that is universal for all autohomeomorphisms of all closed subspaces of $\mathbb{N}^*$. The first result of this paper is that there is no trivial universal autohomeomorphism of $\mathbb{N}^*$, and hence no universal autohomeomorphism at all in any model where all autohomeomorphisms of $\mathbb{N}^*$ are trivial. On the other hand, the Continuum Hypothesis implies that there is a universal autohomeomorphism of $\mathbb{N}^*$. The proof of this will have to be different from the results mentioned above because $\mathbb{N}^*$ is definitely not homeomorphic to its power $(\mathbb{N}^*)^\omega$; it will use group actions and a homeomorphism extension theorem.

We should mention the dual notion of universality where one requires the existence of a surjection $s:X\to Y$ such that $g\circ s=s\circ h$. For the space $\mathbb{N}^*$ this was investigated thoroughly in 2 for general group actions.

1. Some preliminaries

Our notation is standard. For background information on $\mathbb{N}^*$ we refer to 5.

We denote by $\mathsf{Aut}$ the autohomeomorphism group of $\mathbb{N}^*$. We call a member $h$ of $\mathsf{Aut}$ trivial if there are cofinite subsets $A$ and $B$ of $\mathbb{N}$ and a bijection $b:A\to B$ such that $h$ is the restriction of $\beta b$ to $\mathbb{N}^*$.

In both sections we shall use the $G_\delta$-topology on a given space $(X,\tau )$; this is the topology $\tau _\delta$ on $X$ generated by the family of all $G_\delta$-subsets in the given space. It is well-known that $w(X,\tau _\delta )\leqslant w(X,\tau )^{\aleph _0}$; we shall need this estimate in Section 3.

2. What if all autohomeomorphisms are trivial?

To begin we observe that fixed-point sets of trivial autohomeomorphism of $\mathbb{N}^*$ are clopen. Therefore, to show that no trivial autohomeomorphism is universal it would suffice to construct a compact space that can be embedded into $\mathbb{N}^*$ and that has an autohomeomorphism whose fixed-point set is not clopen.

The example

We let $L$ be the ordinal $\omega _1+1$ endowed with its $G_\delta$-topology. Thus all points other than $\omega _1$ are isolated and the neighbourhoods of $\omega _1$ are exactly the co-countable sets that contain it. Then $L$ is a $P$-space of weight $\aleph _1$ and hence, by the methods in 4, Section 2, its Čech-Stone compactification $\beta L$ can be embedded into $\mathbb{N}^*$.

We define $f:L\to L$ such that $\omega _1$ is the only fixed point of $\beta f$. We put

$$\begin{align*} f(\omega _1)&=\omega _1,\\ f(2\cdot \alpha )&=2\cdot \alpha +1, \text{ and }\\ f(2\cdot \alpha +1)&=2\cdot \alpha . \end{align*}$$

This defines a continuous involution on $L$.

If $p\in \beta L\setminus L$ then $p\in \operatorname {cl}\alpha$ for some $\alpha <\omega _1$ and then either $E=\{2\cdot \beta :\beta <\alpha \}$ or $O=\{2\cdot \beta +1:\beta <\alpha \}$ belongs to the ultrafilter $p$. But $f[E]\cap E=\emptyset =f[O]\cap O$, hence $\beta f(p)\neq p$.

Since $\omega _1$ is not an isolated point of $\beta L$, no matter how this space is embedded into $\mathbb{N}^*$ there is no trivial autohomeomorphism of $\mathbb{N}^*$ that would extend $\beta f$.

3. The Continuum Hypothesis

Under the Continuum Hypothesis the space $\mathbb{N}^*$ is generally very well-behaved and one would expect it to have a universal autohomeomorphism as well. We shall prove that this is indeed the case. We need some well-known facts about closed subspaces of $\mathbb{N}^*$.

First we have Theorem 1.4.4 from 5 which characterizes the closed subspaces of $\mathbb{N}^*$ under $\mathsf{CH}$: they are the compact zero-dimensional $F$-spaces of weight $\mathfrak{c}$, and, in addition: every closed subset of $\mathbb{N}^*$ can be re-embedded as a nowhere dense closed $P$-set.

Second we have the homeomorphism extension theorem from 3: $\mathsf{CH}$ implies that every homeomorphism between nowhere dense closed $P$-sets of $\mathbb{N}^*$ can be extended to an autohomeomorphism of $\mathbb{N}^*$.

Step 1.

We consider the natural action of $\mathsf{Aut}$ on $\mathbb{N}^*$; the map $\sigma :\mathsf{Aut}\times \mathbb{N}^*\to \mathbb{N}^*$ given by $\sigma (f,p)=f(p)$. This action is continuous when $\mathsf{Aut}$ carries the compact-open topology $\tau$ and hence also when $\mathsf{Aut}$ carries the $G_\delta$-modification $\tau _\delta$ of $\tau$. For the rest of the construction we consider the topology $\tau _\delta$.

Using this action we define an autohomeomorphism $h:\mathsf{Aut}\times \mathbb{N}^*\to \mathsf{Aut}\times \mathbb{N}^*$ by $h(f,p)=(f,f(p))$. The map $h$ is continuous because its two coordinates are and it is a homeomorphism because its inverse $(f,p)\mapsto (f,f^{-1}(p))$ is continuous as well.

Now if $X$ is a closed subset of $\mathbb{N}^*$ and $g:X\to X$ is an autohomeomorphism then we can re-embed $X$ as a nowhere dense closed $P$-set and we can then find an $f\in \mathsf{Aut}$ such that $f\upharpoonright X=g$. We transfer this embedded copy of $X$ to $\{f\}\times \mathbb{N}^*$ in $\mathsf{Aut}\times \mathbb{N}^*$; for this copy of $X$ we then have $h\upharpoonright X=g$. It follows that $h$ satisfies the universality condition.

Step 2.

We embed $\mathsf{Aut}\times \mathbb{N}^*$ into $\mathbb{N}^*$ in such a way that there is an autohomeomorphism $H$ of $\mathbb{N}^*$ such that $H\upharpoonright (\mathsf{Aut}\times \mathbb{N}^*)=h$. Then $H$ is the desired universal autohomeomorphism of $\mathbb{N}^*$.

To this end we list a few properties of this product.

Weight

The weight of the product is equal to $\mathfrak{c}$, as both factors have weight $\mathfrak{c}$. For $\mathbb{N}^*$ this is clear and for $\mathsf{Aut}$ this follows because the topology $\tau$ has weight $\mathfrak{c}$ and one obtains a base for $\tau _\delta$ by taking the intersections of all countable subfamilies of a base for $\tau$.

Zero-dimensional and $F$

The product is a zero-dimensional $F$-space as the product of the $P$-space $\mathsf{Aut}$ and the compact zero-dimensional $F$-space $\mathbb{N}^*$, see 6, Theorem 6.1.

Strongly zero-dimensional

The product $\mathsf{Aut}\times \mathbb{N}^*$ is not compact, but we shall construct a compactification of it that is also a zero-dimensional $F$-space of weight $\mathfrak{c}$.

For this we need to prove that $\mathsf{Aut}\times \mathbb{N}^*$ is actually strongly zero-dimensional. We prove more: the product is ultraparacompact, meaning that every open cover has a pairwise disjoint open refinement.

Let $\mathcal{U}$ be an open cover of the product consisting of basic clopen rectangles.

For each $f\in \mathsf{Aut}$ there is a finite subfamily $\mathcal{U}_f$ of $\mathcal{U}$ that covers $\{f\}\times \mathbb{N}^*$, say $\mathcal{U}_f=\{C_i\times D_i:i<k_f\}$. Let $C_f=\bigcap _{i<k}C_i$ and $D_{f,i}=D_i\setminus \bigcup _{j<i}D_j$ for $i<k_f$. Then $\mathcal{C}_f=\{C_f\times D_{f,i}:i<k_f\}$ is a disjoint family of clopen rectangles that covers $\{f\}\times \mathbb{N}^*$ and refines $\mathcal{U}$.

Because $\mathsf{Aut}$ has weight $\mathfrak{c}$, and we assume $\mathsf{CH}$, there is a sequence $\langle {f}_\alpha :\alpha \in \omega _{1}\rangle$ in $\mathsf{Aut}$ such that $\{C_{f_\alpha }:\alpha \in \omega _1\}$ covers $\mathsf{Aut}$. Next we let $V_\alpha =C_{f_\alpha }\setminus \bigcup _{\beta <\alpha }C_{f_\beta }$ for all $\alpha$. Because $\mathsf{Aut}$ is a $P$-space the family $\{V_\alpha :\alpha \in \omega _1\}$ is a disjoint open cover of $\mathsf{Aut}$.

The family $\{V_\alpha \times D_{f_\alpha ,i}:i<k_{f_\alpha }, \alpha \in \omega _1\}$ then is a disjoint open refinement of $\mathcal{U}$.

A compactification

To complete Step 2 we construct a compactification of $\mathsf{Aut}\times \mathbb{N}^*$ that is a zero-dimensional $F$-space of weight $\mathfrak{c}$ and that has an autohomeomorphism that extends $h$. The Čech-Stone compactification would be the obvious canditate, were it not for the fact that its weight is equal to $2^\mathfrak{c}$. More precisely, using some continuous onto function from $(\mathsf{Aut},\tau )$ onto $[0,1]$ one obtains a clopen partition of $(\mathsf{Aut},\tau _\delta )$ of cardinality $\mathfrak{c}$. This shows that $\beta (\mathsf{Aut}\times \mathbb{N}^*)$ admits a continuous surjection onto the space $\beta \mathfrak{c}$ (where $\mathfrak{c}$ carries the discrete topology).

To create the desired compactification we build, either by transfinite recursion or by an application of the Löwenheim-Skolem theorem, a subalgebra $\mathbb{B}$ of the algebra of clopen subsets of $\mathsf{Aut}\times \mathbb{N}^*$ that is closed under $h$ and $h^{-1}$, of cardinality $\mathfrak{c}$, and that has the property that for every pair of countable subsets $A$ and $B$ of $\mathbb{B}$ such that $a\cap b=\emptyset$ whenever $a\in A$ and $b\in B$ there is a $c\in \mathbb{B}$ such that $a\subseteq c$ and $c\cap b=\emptyset$ for all $a\in A$ and $b\in B$. The latter condition can be fulfilled because $\mathsf{Aut}\times \mathbb{N}^*$ is an $F$-space — $\bigcup A$ and $\bigcup B$ have disjoint closures — and strongly zero-dimensional — the closures can be separated using a clopen set.

The Stone space $\operatorname {St}(\mathbb{B})$ of $\mathbb{B}$ is then a compactification of $\mathsf{Aut}\times \mathbb{N}^*$ that is a compact zero-dimensional $F$-space of weight $\mathfrak{c}$, with an autohomeomorphism $\bar{h}$ that extends $h$. We embed $\operatorname {St}(\mathbb{B})$ into $\mathbb{N}^*$ as a nowhere dense $P$-set and extend $\bar{h}$ to an autohomeomorphism $H$ of $\mathbb{N}^*$.