Point-countable \(k\)-networks, \(cs^*\)-networks and \(\alpha_4\)-spaces (Q2701862)

This paper considers point-countable \(k\)-networks in terms of \(\alpha_4\)-spaces (i.e., spaces belonging to the class \(\langle 4\rangle\) in the sense of [\textit{A. V. Arkhangel'skij}, Sov. Math., Dokl. 13, 1185-1189 (1972); translation from Dokl. Akad. Nauk SSSR 206, 265-268 (1972; Zbl 0275.54004)]. This paper gives the following results, for example.NEWLINENEWLINENEWLINE(A) Every \(\alpha_4\)-space \(X\) with a point-countable \(k\)-network has a point-countable \(cs^*\)-network \(\mathcal P\) (i.e., for any sequence \(S\) converging to \(x\) and any neighborhood \(U\) of \(x\), there exist \(P\in\mathcal P\) and some subsequence \(T\) of \(S\) such that \(T\cup \{x\}\subset P\subset U)\).NEWLINENEWLINENEWLINE(B) For a \(k\)-space \(X\) with a point-countable \(k\)-network, \(X\) is an \(\alpha_4\)-space \(\Leftrightarrow X\) contains no closed copy of the sequential fan \(S_\omega \Leftrightarrow X\) is \(gf\)-countable (i.e., \(X\) is \(g\)-first countable).