A Stone-Weierstrass type theorem for an unstructured set -- with applications (Q1840745)

By \(F(X)\) (\(F^*(X)\)) the authors denote the topological algebra \(R^X\) (the subalgebra of bounded functions in \(R^X\)) endowed with the topology of uniform convergence. The authors present some general forms of the Stone-Weierstrass theorem. In the most general of these (Theorem 2), no structure is needed on the domain set. It states: Let \(X\) be a set, let \(\mathcal B\) be a subalgebra of \(F^*(X)\), and let \(f\in F^*(X)\). Then \(f\in\text{cl\,}\mathcal B\) iff whenever \(\mathcal F\) is a filter on \(X\) such that \(g(\mathcal F)\) converges for every \(g\in\mathcal B\) then we also have that \(f(\mathcal F)\) converges. As corollaries of their ``no structure'' Stone-Weierstrass theorem, they obtain the version for compact Hausdorff spaces due to M.H. Stone that appears in the Gillman-Jersion text and also a generalization to arbitrary topological spaces due to \textit{L. D. Nel}, [Math. Z. 104, 226--230 (1968; Zbl 0157.29302)]. One difference between the former and Theorem 2 is that Theorem 2 uses stationary filters rather than stationary sets. For the sake of completeness, the classical version of Stone's theorem is derived. Some examples and remarks are given concerning whether or not the requirement that the space \(X\) be compact (or Hausdorff) can be dropped in part or all for each version of the theorem considered. Finally, some Stone-Weierstrass type theorems are given for filter and for Cauchy spaces.