On \(k\)-networks and weak bases for spaces
A family \(\mathcal P\) of subsets of a space \(X\) is a \textit{\(k\)-network} for \(X\) [\textit{P. O'Meara}, Proc. Am. Math. Soc. 29, 183--189 (1971; Zbl 0214.21105)] if for each compact set \(K \subset X\) and an open set \(U \subset X\) containing \(K\), there exists some \(\mathcal Q \in [{\mathcal P}]^{<\omega}\) such that \(K \subset \bigcup \mathcal Q \subset U\). A family \(\mathcal P\) of subsets of a space \(X\) is a \textit{\(cs^*\)-network} for \(X\) [\textit{Z. Gao}, Quest. Answers Gen. Topology 5, 271--279 (1987; Zbl 0636.54026)] if for every sequence \(\{ x_n : n \in \omega \} \subset X\) converging to a point \(x \in X\) and every neighborhood \(U\) of \(x\), there exists some \(P \in \mathcal P\) such that \(x \in P \subset U\) and \(x_n \in P\) for infinitely many \(n \in \omega\). For each point \(x \in X\), let \({\mathcal B}_x\) be a family of subsets of \(X\). Then \({\mathcal B} = \bigcup \{ {\mathcal B}_x : x \in X \}\) is a \textit{weak base} for \(X\) [\textit{A. V. Arhangel'skii}, Russ. Math. Surv. 21, No.~4, 115--162 (1966); translation from Usp. Mat. Nauk 21, No.~4(130), 133--184 (1966; Zbl 0171.43603)] if it satisfies (1) for each \(x \in X\), \(x \in \bigcap {\mathcal B}_x\) and \({\mathcal B}_x\) is closed under finite intersection, and (2) \(G \subset X\) is open if and only if for each \(x \in G\) there exists some \(B \in {\mathcal B}_x\) with \(B \subset G\). A space \(X\) is \textit{weakly first-countable} [\textit{F. Siwiec}, Pac. J. Math. 52, 233--245 (1974; Zbl 0285.54022)] if it has a weak base \({\mathcal B} = \bigcup \{ {\mathcal B}_x : x \in X\}\) such that each \({\mathcal B}_x\) is countable. The main purpose of this paper is to give counterexamples for three problems posed in [\textit{C. Liu} and \textit{Y. Tanaka}, ``Spaces and mappings: special networks'', in: \textit{P. E. Pearl}, Open problems in Topology II. Amsterdam: Elsevier - 23--34 (2007; Zbl 1158.54300)]. Problem 1.4 .Let \(X\) be a Fréchet \(M_3\)-space with a point-countable \(k\)-network. If \(X\) contains no closed copy of \(S_{\omega_1}\), does \(X\) have a point-countable \(cs^*\)-network? Problem 1.5. Is a Fréchet \(M_3\)-space with a point-countable \(k\)-network a Lašnev space? Problem 1.6. Let \(X\) be a weakly first-countable space. Is any weak base for \(X\) a \(k\)-network?
scientific article; zbMATH DE number 5791178
scientific article