If it looks and smells like the reals ...

Given a topological space [X,T] is an element of M, an elementary submodel of set theory, we define X-M to be X boolean AND M with topology generated by {U boolean AND M : U is an element of T boolean AND M}. We prove that if X-M is homeomorphic to R, then X = X-M. The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if "local compactness" is omitted.