Stone-Weierstrass theorems revisited

We prove strengthened and unified forms of vector-valued versions of the Stone-Weierstrass theorem. This is possible by using an appropriate factorization of a topological space, instead of the traditional localizability. Our main Theorem 7 generalizes and unifies number of known results. Applications from the last section include new versions in the scalar case, as well as simultaneous approximation and interpolation under additional constraints.