When is the space \(C_ p(X)\) \(\sigma\)-countably-compact?
For any Tikhonov space X, let \(C_ p(X)\) and \(C^ 0_ p(X)\) be the sets of all (bounded) continuous real-valued functions on X, together with the topology of pointwise convergence. Clearly, \(C_ p(X)\) is \(\sigma\)-compact for any finite X. Further, Arkhangel'skij showed that \(C_ p(X)\) is \(\sigma\)-countably-compact (that is, a countable union of countably compact sets), for any P-space X (that is, a space in which each \(F_{\sigma}\) is closed). The authors establish the converses of both these facts. They also show (again for Tikhonov spaces), that X is finite iff \(C_ p(X)\) is \(\sigma\)-countably-compact, iff \(C_ p(X)\) is \(\sigma\)-compact.
scientific article; zbMATH DE number 4005116