A Stone-Weierstrass Theorem for Polish Spaces, Given by Logic

Abstract

The Stone-Weierstrass theorem obtains every continuous function on a compact space as a uniform limit of functions built from a separating family by finitely many operations, and Reggio proved the theorem equivalent to the Beth definability property of an infinitary equational logic. We extend both results beyond compactness: on every separable metric space, and in particular on every Polish space, every bounded uniformly continuous function is a uniform limit of functions built from countably many bounded separating functions by the connectives of infinitary real-valued logic, and an approximate Beth property of this logic is equivalent to the new theorem. The extension rests on a rule of syntax, which admits a countable supremum when its terms share an a priori bound and thereby makes the sentences define exactly the bounded Lipschitz functions, and on a model existence game that does the work of compactness in the proof of interpolation.