Uniform approximation theorems for real-valued continuous functions

AbstractFor a topological space X, F(X) denotes the algebra of real-valued functions over X and C(X) the subalgebra of all functions in F(X) which are continuous. In this paper we characterize the uniformly dense linear subspaces of C(X) by means of the so-called “Lebesgue chain condition”. This condition is a generalization to the unbounded case of the S-separation by Blasco and Moltó for the bounded case. Through the Lebesgue chain condition we also characterized the linear subspaces of F(X) whose uniform closure is closed under composition with uniformly continuous functions