If it looks and smells like the reals...

Abstract

Given a topological space ∈ M, an elementary submodel of set theory, we define XM to be X ∩ M with topology generated by {U ∩ M : U ∈ T ∩ M}. We prove that if XM is homeomorphic to R, then X = XM . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted. Given a model of set theory, i.e. a collection W of sets which satisfies the usual set-theoretic axioms (ZFC), a set M ⊆ W is an elementary submodel of W if for every natural number n and for every formula φ with n free variables in the predicate calculus with = and a 2-place relation symbol ∈, and every x1, · · · , xn ∈ M (we will systematically confuse the membership relation and the symbol ’∈’), φ(x1, · · · , xn) holds in M if and only if it does in W. We usually think of W as being V, the universe of all sets, but for technical reasons officially deal with W = H(θ), the collection of all sets of hereditary cardinality less than θ, a “sufficiently large” regular uncountable cardinal and rather than dealing with ZFC, we deal with sufficiently large fragments of it. (For more on these technical reasons, see [JW].) The non-logician reader will not lose much by thinking of elementary submodels of V. Elementary submodels have been used in set-theoretic topology with increasing frequency and depth over the past 20 years (see e.g. [D]). As often happens in mathematics, one’s tools become objects of study; thus in [JT] we inaugurated a systematic investigation of the topological spaces induced by elementary submodels. This paper is a continuation of that study, although it is mainly independent of [JT]. The Downward Lowenheim-Skolem Theorem of Logic implies that, given any set X ∈ H(θ) and an infinite cardinal κ ≤ |H(θ)| , there is an elementary submodel M of H(θ) with X ∈ M and |M | = κ. Given a topological space ∈ M, we define XM to be the space X∩M with topology TM generated by {U ∩M : U ∈ T ∩M}. The Downward Lowenheim-Skolem Theorem yields XM ’s with X ∩ M having any infinite cardinality ≤ |X | ; a natural question is whether an Upward Lowenheim-Skolem Theorem holds in this context, i.e. 1Research supported by NSERC grant A-7354. AMS Mathematics Subject Classification. Primary 03C62, 03E35, 54A35. Secondary 54F65.