Universal autohomeomorphisms of

By Klaas Pieter Hart and Jan van Mill

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


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.


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 -topology on a given space ; this is the topology  on  generated by the family of all -subsets in the given space. It is well-known that ; 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 -topology. Thus all points other than  are isolated and the neighbourhoods of  are exactly the co-countable sets that contain it. Then is a -space of weight  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 -spaces of weight , 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 -sets of  can be extended to an autohomeomorphism of .

Step 1.

We consider the natural action of on ; the map given by . This action is continuous when carries the compact-open topology  and hence also when carries the -modification of . For the rest of the construction we consider the topology .

Using this action we define an autohomeomorphism by . The map  is continuous because its two coordinates are and it is a homeomorphism because its inverse is continuous as well.

Now if is a closed subset of  and is an autohomeomorphism then we can re-embed as a nowhere dense closed -set and we can then find an such that . We transfer this embedded copy of  to  in ; for this copy of  we then have . It follows that satisfies the universality condition.

Step 2.

We embed into  in such a way that there is an autohomeomorphism  of  such that . Then is the desired universal autohomeomorphism of .

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


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 -space as the product of the -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 -space of weight .

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 -space the family 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 -space of weight  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 -space of weight , with an autohomeomorphism  that extends . We embed into  as a nowhere dense -set and extend  to an autohomeomorphism  of .

Mathematical Fragments

Step 2.

We embed into  in such a way that there is an autohomeomorphism  of  such that . Then is the desired universal autohomeomorphism of .


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P. C. Baayen, Universal morphisms, Mathematical Centre Tracts, vol. 9, Mathematisch Centrum, Amsterdam, 1964. MR0172826,
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Article Information

MSC 2020
Primary: 54D40 (Remainders in general topology)
Secondary: 03E50 (Continuum hypothesis and Martin’s axiom), 54A35 (Consistency and independence results in general topology)
  • Autohomeomorphism
  • universality
Author Information
Klaas Pieter Hart
Faculty EEMCS, TU Delft, Postbus 5031, 2600 GA Delft, the Netherlands
Jan van Mill
KdV Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands
Communicated by
Vera Fischer
Journal Information
Proceedings of the American Mathematical Society, Series B, Volume 9, Issue 8, ISSN 2330-1511, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , and published on .
Copyright Information
Copyright 2022 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
Article References
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  • DOI 10.1090/bproc/106
  • MathSciNet Review: 4398473
  • Show rawAMSref \bib{4398473}{article}{ author={Hart, Klaas Pieter}, author={van Mill, Jan}, title={Universal autohomeomorphisms of $\mathbb{N}^*$}, journal={Proc. Amer. Math. Soc. Ser. B}, volume={9}, number={8}, date={2022}, pages={71-74}, issn={2330-1511}, review={4398473}, doi={10.1090/bproc/106}, }


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