If $X$ is a finite CW complex then according to Milnor's article On Spaces Having the Homotopy Type of a CW-Complex, the mapping space
$$Map(X,X)$$
(which we furnish with the compact-open topology) has the homotopy type of a CW complex. Of course Milnor's statement is more general, but this is all I need to know for this.
The composition of maps is continous and makes $Map(X,X)$ into a topological monoid. When we consider the subset of invertible elements
$$Homeo(X)\subseteq Map(X,X)$$
consisting of all the self-homeomorphisms of $X$, the assumptions on $X$ make inversion continuous and $Homeo(X)$ becomes a topological group.
Is $Homeo(X)$ a subcomplex of $Map(X,X)$? Or more generally does $Homeo(X)$ admit a CW structure (possibly unrelated to that of $Map(X,X)$)?
This should be well-known but I haven't been able to find a reference so I decided to pitch it to you guys. There is enough in the literature covering the infinite-dimensional manifold structure on $Diff(X)$ when $X$ is a compact, smooth manifold, but I haven't been able to dig anything up on the more general homeomorphism groups.