# Universal autohomeomorphisms of

To the memory of Cor Baayen, who taught us many things

## Abstract

We study the existence of universal autohomeomorphisms of We prove that the Continuum Hypothesis .( implies there is such an autohomeomorphism and show that there are none in any model where all autohomeomorphisms of ) are trivial.

## Introduction

This paper is concerned with universal autohomeomorphisms on the Čech-Stone remainder of , .

In very general terms we say that an autohomeomorphism on a space is *universal* for a class of pairs where , is a space and is an autohomeomorphism of if for every such pair there is an embedding , such that that is, , extends the copy of on .

In Reference 1, Section 3.4 one finds a general way of finding universal autohomeomorphisms. If is homeomorphic to then the shift mapping defines a universal autohomeomorphism for the class of all pairs where , is a subspace of One embeds . into by mapping each to the sequence .

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 carries an autohomeomorphism that is universal for all autohomeomorphisms of completely regular spaces of weight at most and the Cantor cube , has a universal autohomeomorphism for all zero-dimensional such spaces.

Our goal is to have an autohomeomorphism on that is universal for all autohomeomorphisms of all *closed* subspaces of The first result of this paper is that there is no trivial universal autohomeomorphism of . and hence no universal autohomeomorphism at all in any model where all autohomeomorphisms of , are trivial. On the other hand, the Continuum Hypothesis implies that there is a universal autohomeomorphism of The proof of this will have to be different from the results mentioned above because . is definitely not homeomorphic to its power 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 such that For the space . this was investigated thoroughly in Reference 2 for general group actions.

## 1. Some preliminaries

Our notation is standard. For background information on we refer to Reference 5.

We denote by the autohomeomorphism group of We call a member . of *trivial* if there are cofinite subsets and of and a bijection such that is the restriction of to .

In both sections we shall use the on a given space -topology this is the topology ; on generated by the family of all in the given space. It is well-known that -subsets 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 are clopen. Therefore, to show that no trivial autohomeomorphism is universal it would suffice to construct a compact space that can be embedded into and that has an autohomeomorphism whose fixed-point set is not clopen.

### The example

We let be the ordinal endowed with its Thus all points other than -topology. are isolated and the neighbourhoods of are exactly the co-countable sets that contain it. Then is a of weight -space and hence, by the methods in Reference 4, Section 2, its Čech-Stone compactification can be embedded into .

We define such that is the only fixed point of We put .

This defines a continuous involution on .

If then for some and then either or belongs to the ultrafilter But . hence , .

Since is not an isolated point of no matter how this space is embedded into , there is no trivial autohomeomorphism of that would extend .

## 3. The Continuum Hypothesis

Under the Continuum Hypothesis the space 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 .

First we have Theorem 1.4.4 from Reference 5 which characterizes the closed subspaces of under they are the compact zero-dimensional : of weight -spaces and, in addition: every closed subset of , can be re-embedded as a nowhere dense closed -set.

Second we have the homeomorphism extension theorem from Reference 3: implies that every homeomorphism between nowhere dense closed of -sets can be extended to an autohomeomorphism of .

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

#### Weight

The weight of the product is equal to as both factors have weight , For . this is clear and for this follows because the topology has weight and one obtains a base for by taking the intersections of all countable subfamilies of a base for .

#### Zero-dimensional and

The product is a zero-dimensional as the product of the -space -space and the compact zero-dimensional -space see ,Reference 6, Theorem 6.1.

#### Strongly zero-dimensional

The product is not compact, but we shall construct a compactification of it that is also a zero-dimensional of weight -space .

For this we need to prove that is actually strongly zero-dimensional. We prove more: the product is ultraparacompact, meaning that every open cover has a *pairwise disjoint* open refinement.

Let be an open cover of the product consisting of basic clopen rectangles.

For each there is a finite subfamily of that covers say , Let . and for Then . is a disjoint family of clopen rectangles that covers and refines .

Because has weight and we assume , there is a sequence , in such that covers Next we let . for all Because . is a the family -space is a disjoint open cover of .

The family then is a disjoint open refinement of .

#### A compactification

To complete Step 2 we construct a compactification of that is a zero-dimensional of weight -space and that has an autohomeomorphism that extends The Čech-Stone compactification would be the obvious canditate, were it not for the fact that its weight is equal to . More precisely, using some continuous onto function from . onto one obtains a clopen partition of of cardinality This shows that . admits a continuous surjection onto the space (where 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 of the algebra of clopen subsets of that is closed under and of cardinality , and that has the property that for every pair of countable subsets , and of such that whenever and there is a such that and for all and The latter condition can be fulfilled because . is an — -space and have disjoint closures — and strongly zero-dimensional — the closures can be separated using a clopen set.

The Stone space of is then a compactification of that is a compact zero-dimensional of weight -space with an autohomeomorphism , that extends We embed . into as a nowhere dense and extend -set to an autohomeomorphism of .