Long colimits of topological groups IV: Spaces with socks

TL;DR

This paper investigates the topological properties of spaces with the Compactly Supported Homeomorphism Property (CSHP), demonstrating that certain product spaces like the Long Line also possess CSHP, thus advancing understanding of their topological group structures.

Contribution

The authors develop new techniques to show that products of spaces with CSHP, such as the Long Line, also have CSHP, extending previous results in topological group theory.

Findings

Spaces with CSHP include the Long Line and the Closed Long Ray.

Products of spaces with CSHP also have CSHP.

Techniques for verifying CSHP in product spaces are established.

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 develop techniques for showing that products of certain spaces with CSHP, such as the Closed Long Ray and the Long Line, have CSHP again.