On groups of compactly supported homeomorphisms

Given a compact space , it is well known that the homeomorphism group is a topological group with the compact-open topology. If is a Tychonoff space and is compact, then the group of homeomorphisms supported in  is a topological group with the compact-open topology; it also embeds as a subgroup into .
The group of compactly supported homeomorphisms of can be topologized in more than one way. can be equipped with the compact-open topology induced by ; however, it can also be equipped with the finest topology making all inclusions continuous. When these two topologies coincide, we say that has the Compactly Supported Homeomorphism Property (CSHP).
In this talk, we present results toward characterizing spaces that have CSHP. We provide necessary and sufficient conditions for finite products of ordinals, necessary conditions for the finite product and coproduct (i.e., disjoint union) of locally compact spaces with compact spaces, and sufficient conditions in terms of the structure of the space’s compact subsets.
Joint work with Rafael Dahmen.