Long colimits of topological groups III: Homeomorphisms of products and coproducts

TL;DR

This paper investigates conditions under which the group of compactly supported homeomorphisms on certain spaces, especially products of ordinals, have a specific topological property called CSHP, linking different topologies on these groups.

Contribution

It provides necessary and sufficient conditions for finite products of ordinals to possess the CSHP, advancing understanding of topological homeomorphism groups.

Findings

Finite products of ordinals with the order topology can have CSHP under specific conditions.

Necessary conditions are identified for finite products and coproducts to have CSHP.

The paper clarifies when different topologies on homeomorphism groups coincide.

Abstract

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support, or as a subgroup of the homeomorphism group of its Stone-\v{C}ech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.