Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ 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.