Basis-to-space transfer for Homp(X) being a topological group

Determine whether Homp(X), the group of homeomorphisms of X with the topology of point-wise convergence, is a topological group under the assumption that X has a basis B such that Homp(U) is a topological group for every basis element U in B.

Background

Following several positive results that ensure Homp(X) is a topological group for particular constructions (e.g., Alexandroff Noodles) or under local uniqueness conditions, the author poses local-to-global questions. The first asks whether having the property on all basis elements is sufficient to ensure the property for the whole space.

This addresses a potential method for constructing new classes of spaces with Homp(X) a topological group by verifying the property on simpler local pieces.