Topological spaces with the Freese-nation property. II (Q2665204)

This article is a continuation of the research of \textit{J. Bak} and \textit{A. Kucharski} [Ann. Math. Sil. 33, 41--54 (2019; Zbl 1429.54018)] where the FN-property and the FNS-property were introduced. Here the authors study, moreover, the \(\pi\)-FNS property. A topological space has the \(\pi\)-FNS property if there exists a \(\pi\)-base with the FNS property. It is proved, in particular, that compact Hausdorff spaces with the FNS property for some base consisting of co-zero sets are openly generated spaces, and spaces with the \(\pi\)-FNS property are skeletally generated. Moreover, examples of families without the FN property and without the FNS property are given.