Countable normality of subsets of \(\exp (X)\)
A space is called countably normal (pseudonormal, respectively) if any two closed disjoint sets such that one of them is countable and discrete (only countable, respectively) are contained in disjoint open sets. The countable normality was defined in 1923 by \textit{P. S. Alexandroff} and \textit{P. Urysohn} [Verhandelingen Amsterdam 14, No. 1 (1929; JFM 55.0960.02)] and much later it was defined under the name \(D\)-property by \textit{E. K. van Douwen} [The integers and topology, in `Handbook of set-theoretic topology', 111-167 (1984; Zbl 0561.54004)]. Pseudonormal spaces were introduced in [\textit{C. W. Proctor}, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 18, 179-181 (1970; Zbl 0191.21102)]. The following relations are obvious: normality \(\Rightarrow\) pseudonormality \(\Rightarrow\) countable normality. The space of closed subsets of the space \(X\) with Vietoris topology is denoted by \(\exp(X)\). The hereditary pseudonormality of \(\exp(X)\) implies the countable normality, perfect normality, and hereditary separability of \(X\) [the author, Mosc. Univ. Math. Bull. 47, No. 1, 48-50 (1992); translation from Vestn. Mosk. Univ., Ser. I 1992, No. 1, 102-104 (1992; Zbl 0764.54019)]. The following theorem is an amplification of the latter statement: Theorem 1: If \(\exp(X)\) is a hereditarily countable normal space, then the space \(X\) is countably compact, perfectly normal, and hereditarily separable.
scientific article; zbMATH DE number 1256314