Calibers of spaces of functions and the metrization problem for compact subsets of \(C_ p(X)\) (Q1107867)

C\({}_ p(X)\) is the space of continuous realvalued maps endowed with the pointwise convergence (X is completely regular), \(C_ p(C_ p(X))\) is denoted by \(C_{p,2}(X)\), \(L_ p(X)\) is the linear hull of X in \(C_{p,2}(X)\). The main result states that a regular cardinal is a caliber of X iff it is a caliber of \(L_ p(X)\) iff it is a caliber of \(C_{p,2}(X)\). One of many interesting consequences asserts that the reviewer's conjecture (compact space with \(\omega_ 1\)-inaccessible diagonal is metrizable [Comment. Math. Univ. Carolinae 18, 777-788 (1977; Zbl 0374.54035)]) is equivalent to the assertion that if \(\omega_ 1\) is a caliber of X, then every compact set in \(C_ p(X)\) is metrizable. By the result of \textit{H. Zhou} [Top. Appl. 13, 283-293 (1982; Zbl 0495.54028)], it is true under CH-FA; in ZFC it is shown here that the result is true if, moreover, X contains a dense continuous image of a product of separable spaces (then even pseudocompact subspaces of \(C_ p(X)\) are metrizable). Using a pseudocompact space X having caliber \(\omega_ 1\) and the property that any its countable subspace is closed and C *-embedded (constructed by D. V. Šachmatov), it is shown that \(C_ p(X,[0,1])\subset C_ p(X)\) is a pseudocompact subspace without \(G_{\delta}\)-diagonal.