On normality in function spaces
We present some results for the following general problem: if \(C_p(X)\) is normal, then what properties must \(X\) have? Here and in what follows \(C_p(X)\) is the space of all continuous functions on \(X\) in a pointwise convergence topology. For the above-mentioned problem, the first result obtained is that if \(C_p(X)\) is Lindelöf and \(X\) is a compact set, then \(X\) has a denumerable tightness [\textit{H. H. Corson}, Trans. Am. Math. Soc. 101, 1-15 (1961; Zbl 0104.08502)]. After that the following results were obtained: \textit{M. O. Asanov} [Modern topology and set theory, No. 2, 8-12 (1979; Zbl 0484.54006)] established that the Lindelöf property of \(C_p (X)\) implies that \(X\) has a denumerable tightness of any finite degree; here \(X\) is not assumed to be compact. In [Russ. Math. Surv. 37, No. 4, 149-150 (1982); translation from Usp. Mat. Nauk 37, No. 4(226), 149-150 (1982; Zbl 0527.54013)] the author proved that if \(C_p(X)\) is normal, then any compact set of \(X\) has a denumerable tightness. Finally, \textit{E. A. Reznichenko} and \textit{D. P. Baturov} [\textit{A. V. Arhangel'skij}, Topological function spaces (1989; Zbl 0781.54014), p. 139] obtained the following remarkable result: if \(X\) is compact (or, more generally, a Lindelöf \(\Sigma\)-space), then the Lindelöf property and normality are equivalent in \(C_p(X)\). In this paper, we leave the class of compact sets and consider its nearest vicinity. We also study the influence of the normality of \(C_p(X)\) on the behavior of denumerable-compact, \(\omega_1\)-compact, and pseudo-compact sets. We establish the following qualitative results. (1) \(\omega_1\)-compact sets are homothetic to compact sets, i.e., they have denumerable tightness. (2) Denumerable-compact sets lose this property, but possess another curious property: such sets cannot contain points of tightness \(\omega_1\). (3) If \(X\) is pseudo-compact, then the normality of \(C_p(X)\) implies that the Lindelöf number of \(C_p(X)\) cannot be equal to \(\omega_1\). In particular, for a pseudo-compact set \(X\) of network weight \(\leq\omega_1\), \(C_p(X)\) is Lindelöf if and only if it is normal.
scientific article; zbMATH DE number 830751