Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

TL;DR

Under the Axiom of Determinacy, the paper proves that all zero-dimensional homogeneous spaces are strongly homogeneous, except for locally compact ones, extending previous results and addressing open questions in topology.

Contribution

It establishes that, assuming determinacy, zero-dimensional homogeneous spaces are strongly homogeneous, generalizing prior Borel space results and partially answering longstanding questions.

Findings

All zero-dimensional homogeneous spaces are strongly homogeneous under determinacy.

The result applies to spaces generating a given Wadge class.

It extends previous work by van Engelen and addresses questions by Terada and Medvedev.

Abstract

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (that is, all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.