Spaces hereditarily of \(\kappa\)-type and point \(\kappa\)-type (Q1329609)

For a space \(X\), and set \(C\subset X\), the character of \(C\) in \(X\) is defined to be the smallest infinite cardinal \(\kappa\) such that there is a family \({\mathcal B}\) of open sets with \(| {\mathcal B}|\leq \kappa\) such that for every open \(U\supset C\) there exists \(B\in {\mathcal B}\) such that \(C\subset B\subset U\). A space \(X\) is said to have point \(\kappa\)- type provided every point in \(X\) is contained in a compact set \(C\) of character at most \(\kappa\). A space is said to have \(\kappa\)-type if every compact set is contained in a compact set of character at most \(\kappa\). Point \(\omega\)-type was introduced by \textit{A. V. Arkhangel'skij} [Soviet Math., Dokl. 4, 1051-1055 (1968); translation from Dokl. Akad. Nauk SSSR 151, 751-754 (1963; Zbl 0124.158)], and \(\omega\)-type was introduced by \textit{M. Henriksen} and \textit{J. R. Isbell} under a different name [Duke Math. J. 25, 83-105 (1958; Zbl 0081.386)]. Among other results, the authors prove that a regular space \(X\) is hereditarily of point \(\kappa\)-type if and only if for every \(p\in X\) there is a set \(E_ p\) (possibly empty) of isolated points of \(X\) such that \(E_ p\cup \{p\}\) is a compact set of character at most \(\kappa\). An analogous result is obtained for regular spaces hereditarily of \(\kappa\)-type.