Approximation by pointwise bounded sets of continuous functions

Abstract

Assuming that X is a Tychonoff space, we obtain necessary and sufficient conditions for a function f∈RX to belong to the closure in RX of a pointwise bounded family F in C(X), the ring of real-valued continuous functions on X. In other words, if M(X) stands for the bidual of the space Cp(X), equipped with the pointwise topology, we provide necessary and sufficient conditions for a function f∈RX to belong to M(X). As a noteworthy fact, if M, S and R are, respectively, the Michael, the Sorgenfrey, and the real line, it is shown that C(M)⊈M(R) but C(S)⊆M(R).