If it looks and smells like the reals...
Volume 163 / 2000
Fundamenta Mathematicae 163 (2000), 1-11
DOI: 10.4064/fm-163-1-1-11
Abstract
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.
