Projective \(\sigma\)-compactness, \(\omega_1\)-caliber, and \(C_p\)-spaces
Let \(C_p(X)\) be the space of continuous real-valued functions on \(X\) with the topology of pointwise convergence. A space is projectively \(\sigma\)-compact provided that every separable metrizable continuous image of the space is \(\sigma\)-compact. This article shows that \(C_p(X)\) is projectively \(\sigma\)-compact if and only if it is \(\sigma\)-pseudocompact. In addition, characterizations are given for \(C_p(X)\) to be projectively analytic and for \(C_p(Y|X)\) to be \(\sigma\)-compact, where \(C_p(Y|X)\) consists of the restrictions of the functions in \(C_p(X)\) to \(Y\). In the last section, a known result concerning spaces with cardinality no greater than \(2^{\omega}\) is generalized and used to show that the weight of a compact subspace of \(C_p(X)\) is no greater than \(2^{\omega}\) whenever \(\omega_1\) is a caliber of \(X\) and \(X\) has countable extent. This is a ZFC result so that it is related to the question of whether it is true in ZFC that every compact subspace of \(C_p(X)\) is metrizable whenever \(\omega_1\) is a caliber of \(X\).
scientific article; zbMATH DE number 1462690