A Stone-Weierstrass Type Theorem for an Unstructured Set - with Applications

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A general version of the Stone–Weierstrass theorem is presented – one which involves no structure on the domain set of the real valued functions. This theorem is similar to the ‘Stone–Weierstrass theorem’ which appears in the book by Gillman and Jerison, but instead of involving the concept of stationary sets the one presented here involves stationary filters. As a corollary to our results we obtain Nel's theorem of Stone–Weierstrass type for an arbitrary topological space. Finally, an application is made to the setting of Cauchy spaces.