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3Rcp algebraic assume axiomatizable axiomatized axioms Choose clopen sets closed under countable compactness theorem countable intersections cp iff cp(x denote dense denumerable euclidean eurd Exercise following are equivalent Given hausdorff Hence hereditarily undecidable holds homeomorphic implies induction infinite interpolation theorem invariant for topologies isomorphism L-structure L^-sentences Lemma Let A,a locally bounded locally bounded real locally pure logic Lowenheim-Skolem theorem many-sorted model of cp model theory monotone structures monotone system negation normal form neighborhood obtain order topology partial homeomorphism Proof prove proximity spaces quantifiers recursively saturated relation resp sentences preserved sequence set variables si,a struc subset subspace substructure Suppose T„-spaces topological field topological groups topological spaces topological structures topological vector space torsionfree two-sorted uniform spaces V-topology valuation ring VX 3Y weak structure