Completeness theorems for continuous functions and product topologies
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In this paper we formulate a first order theory of continuous functions on product topologies via generalized quantifiers. We present an axiom system for continuous functions on product topologies and prove a completeness theorem for them with respect to topological models. We also show that if a theory has a topological model which satisfies the Hausdorff separation axiom, then it has a 0-dimensional, normal topological model. We conclude by obtaining an axiomatization for topological algebraic structures, e.g. topological groups, proving a completeness theorem for the analogue with countable conjunctions and disjunctions, and presenting counterexamples to interpolation and definability.
KeywordsTopological Space Topological Group Topological Model Product Topology Completeness Theorem
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