A variation on the Stone-Weierstrass theorem

Abstract

Giving Ix the topology of uniform convergence, we have that the closure of a set with property V has property V, as does the intersection of such sets. Thus every subset of Ix is contained in a smallest set with property V, and in a smallest closed set with property V. If X is a topological space then the set D(X) of all continuous functions from X into I is closed and has property V. The idea of considering such collections of functions comes from a statement of von Neumann in [1]. Essentially, he claims without proof what we give here as a corollary to Theorem 2. I am indebted to Dr. R. S. Pierce for bringing the problem to my attention. DEFINITION. If n is a positive integer, let P,n be the smallest subset of D(In) that has property V and contains the n projections.