Existence of nice resolutions in Cp(X) and its bidual often implies metrizability of Cp(X)

Abstract Let X be a Tychonoff space. We show that C p ( X ) , the space of real-valued continuous functions on X equipped with the pointwise topology, admits a resolution (an ordered covering indexed by N N ) consisting of convex compact sets that swallows the local null sequences if and only if X is countable and discrete. Then we prove (main result) (i) the weak* bidual of C p ( X ) admits a resolution consisting of bounded sets if and only if X is countable. Hence ( i i ) if C p ( X ) admits a fundamental bounded resolution, X must be countable [9] . If X is first countable, then C p ( X ) admits a resolution made up of bounded sets swallowing the Cauchy sequences if and only if X is countable. In the context of the present research, the weak* bidual of C p ( X ) has received little or null attention so far. Result (i) fixes this situation.