Some sufficiency conditions for \(D\)-spaces, dually discrete and its applications
Let \(X\) be a topological space and for each \(x\in X\) let \(\mathcal{W}(x)\) be a family of subsets of \(X\) containing \(x\). In [\textit{P. J. Collins} and \textit{A. W. Roscoe}, Proc. Am. Math. Soc. 90, 631--640 (1984; Zbl 0541.54034)], the following definition was given: \(\mathcal{W}=\{\mathcal{W}(x):x\in X\}\) satisfies the property \((F)\) if for every \(x\in X\) and every open neighborhood \(U\) of \(x\) there exists an open set \(V\) containing \(x\) such that \(x\in W\subseteq U\) for some \(W\in \mathcal{W}(y)\) whenever \(y\in V\). If \(\mathcal{P}\) is a property, then \(\mathcal{W}\) satisfies \(\mathcal{P}(F)\) if \(\mathcal{W}\) satisfies \((F)\) and for each \(x\in X\), \(\mathcal{W}(x)\) has the property \(\mathcal{P}\). We recall that a \textit{neighborhood assignment} in \(X\) is a family of open sets \(\{O_x:x\in X\}\) such that \(x\in O_x\) for every \(x\in X\). Using special neighborhood assignments or families \(\mathcal{W}\) satisfying \(\mathcal{P}(F)\) for \(\mathcal{P}\in\) \{well-ordered, chain, chain neighborhood, closure-preserving chain, \(\omega_1\)-sheltering, \(\omega_1\)-sheltering chain, etc.\}, the author of this paper proves several sufficiency conditions for a topological space \(X\) to be a \(D\)-space (paracompact \(D\)-space, hereditary \(D\)-space), dually discrete space or dually scattered space of rank \(\leq 2\). As corollaries of these theorems, the author obtains some previously known results. Several examples are given.
dually discrete \(\kappa\)-sheltering \((F)\) space
scientific article; zbMATH DE number 6672884
scientific article