A study on monotonically metacompact and property \((A)((B))\)
A topological space \(X\) is \textit{(countably) metacompact} if each (countable) open cover of \(X\) has a point-finite open refinement which also covers \(X\), and \(X\) is said to be \textit{monotonically (countably) metacompact} if there is a function \(r\) that associates with each (countable) open cover \(\mathcal U\) of \(X\) a point-finite open refinement \(r(\mathcal U)\) that also covers \(X\) in such a way that the following statement holds: if \(\mathcal U\) and \(\mathcal{V}\) are open covers of \(X\), then \(r(\mathcal U)\) refines \(r(\mathcal V)\) whenever \(\mathcal U\) refines \(\mathcal V\). A generalized ordered topological space (GO-space) is a triple \((X,\tau,<)\), where \(<\) is a linear ordering on \(X\) and \(\tau\) is a Hausdorff topology on \(X\) which is generated by a base of order-convex sets. It \(\tau\) is the open interval topology of the given ordering, then \((X,\tau,<)\) is a linearly ordered topological space (LOTS). Properties (A) and (B) where introduced by \textit{T. Chase} and \textit{G. Gruenhage} [Topology Appl. 213, 135--144 (2016; Zbl 1356.54031)]. Let \(X\) be a topological space and \(P(X)\) be the collection of all pairs \((x,U)\) where \(x \in X\) and \(U\) is an open neighbourhood of \(x\). A space has property (A) (resp. property (B)) if to each \((x,U) \in P(X)\) one can assign an open set \(V(x,U)\) such that \(x \in V(x,U) \subseteq U\) and for any collection \(\{(x_\alpha,U_\alpha): \alpha \in A\} \subseteq P(X)\), either \(\bigcap\limits_{\alpha \in A} V(x_\alpha,U_\alpha) = \emptyset\) or there is a finite \(A' \subseteq A\) such that for each \(\alpha \in A\) there is some \(\beta \in A'\) with \(V(x_\alpha,U_\alpha) \subseteq U_\beta\) (resp. \(\{x_\alpha: \alpha \in A\} \subseteq \bigcup\limits_{\beta \in A'} U_\beta\)). Chase and Gruenhage have shown, in the paper referred to, that compact spaces which satisfy property (A) are metrizable, and a separable space which satisfies property (A) is metrizable. In the paper under review, the authors show that if a regular topological space has \textit{caliber \(\omega_1\)} (i.e., for every uncountable collection \(\mathcal U\) of non-empty sets of \(X\) there is an uncountable subcollection \(\mathcal V \subseteq \mathcal U\) such that \(\bigcap \mathcal V \neq \emptyset\)) and satisfies property (A), then \(X\) is metrizable. It was previously shown in [\textit{H. Li} and \textit{L. Peng}, J. Math. Res. Appl. 33, No. 3, 353--360 (2013; Zbl 1289.54081)] that a GO-space \(X\) is monotonically metacompact if and only if its closed linearly ordered extension \(X^*\) is monotonically metacompact. In the paper under review, the authors show that if \(X\) is a GO-space with property (A) (resp. (B)) then the closed linearly ordered extension \(X^*\) of \(X\) has property (A) (resp. (B)). A number of additional results are established. For instance, it is shown in the paper that if \(X\) is a scattered GO-space which has property (A) then \(X\) is monotonically metacompact.
scientific article; zbMATH DE number 6910242
scientific article