Ferrando, J.C. (2022) «Approximation by pointwise bounded sets of continuous functions», Topology and its Applications, 317:108191

Juan Carlos Ferrando (Operations Research Center, University Miguel Hernández of Elche) 

Abstract: Assuming that X is a Tychonoff space, we obtain necessary and sufficient conditions for a function f∈R^Xto belong to the closure in R^Xof 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 C_p(X), equipped with the pointwise topology, we provide necessary and sufficient conditions for a function f∈R^Xto 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).