On some applications of property (A) ((\(\sigma\)-A)) at a point (Q6592030)

For any topological space \(X\), the family \(\tau(X)\) is its topology and \(\tau(x,X)=\{U\in \tau(X): x\in U\}\). Given a space \(X\), let \(\mathcal T_X= \{(x,U): x\in U \in \tau(X)\}\). A space \(X\) has the property \((A)\) if to each \((x,U) \in \mathcal T_X\), one can assign an open set \(V(x,U)\) such that \(x\in V(x,U)\subset U\) and for any family \(\{(x_\alpha, U_\alpha): \alpha\in A\}\subset \mathcal T_X\), either \(\bigcap_{\alpha\in A} V(x_\alpha,U_\alpha)=\emptyset\) or there exists a finite set \(A'\subset A\) such that for every \(\alpha\in A\) we have the inclusion \(V(x_\alpha, U_\alpha) \subset U_\beta\) for some \(\beta\in A'\). The space \(X\) has the property \((\sigma\)-\(A)\) if to each \((x,U)\in \mathcal T_X\) one can assign an open set \(V(x,U)\) such that \(x\in V(x,U)\subset U\) and for any family \(\{(x_\alpha, U_\alpha): \alpha\in A\}\subset \mathcal T_X\), either \(\bigcap_{\alpha\in A} V(x_\alpha,U_\alpha)=\emptyset\) or there exists a family \(\{A_n: n\in \omega\}\) such that \(A= \bigcup\{A_n: n\in \omega\}\) and, given any \(n\in\omega\) and a set \(A'_n\subset A_n\), there exists a finite set \(A''_n\subset A'_n\) such that, for each \(\alpha\in A_n'\) we have the inclusion \(V(x_\alpha,U_\alpha) \subset U_\beta\) for some \(\beta\in A_n''\).\N\NIf \(x\in X\) and, in the above definitions of the properties \((A)\) and \((\sigma\)-\(A)\), instead of \(\mathcal T_X\), we consider the family \(\mathcal T_x\) of all the pairs \((x,U) \in \mathcal T_X\), then we obtain the definitions of the properties \((A)\) and \((\sigma\)-\(A)\) at the point \(x\). A family \(\mathcal B\) of pairs \((B_1,B_2)\) is a pair-base in \(X\) if \(B_1\in \tau(X)\) and \(B_1\subset B_2\) for each \((B_1,B_2)\in \mathcal B\) and for every \(x\in X\) and \(U\in \tau(x,X)\), there exists \((B_1,B_2)\in \mathcal B\) such that \(x\in B_1 \subset B_2\subset U\). A family \(\mathcal A\) of pairs \((A_1,A_2)\) of subsets of \(X\) is an NSR pair-family if \(A_1 \subset A_2\) for every \((A_1,A_2)\in \mathcal A\) and for any \(\mathcal A' \subset \mathcal A\) such that \(\bigcap\{A_1: (A_1,A_2)\in \mathcal A'\} \neq \emptyset\), there is a finite family \(\mathcal F\subset \mathcal A'\) such that, for every \((A_1,A_2)\in \mathcal A'\), we have the inclusion \(A_1\subset F_2\) for some \((F_1,F_2)\in \mathcal F\). A pair-base \(\mathcal B\) is called a \(\sigma\)-NSR pair-base if \(\mathcal B\) is a countable union of NSR pair-families. If, in the definition of a pair-base \(\mathcal B\) in \(X\), we only consider a given point \(x\in X\), then \(\mathcal B\) is called a pair-base at the point \(x\).\N\NA space \(X\) is \(\omega\)-scattered if every non-empty subspace of \(X\) has a non-empty relatively open countable subspace. It is established in the paper that if \(X\) is a hereditarily metacompact \(\omega\)-scattered space and \(X\) has a \(\sigma\)-NSR pair-base at every point, then \(X\) has a \(\sigma\)-NSR pair-base. If \(X\) is a hereditarily meta-Lindelöf \(\omega\)-scattered space, and \(X\) has a \(\sigma\)-NSR pair-base at every point, then \(X\) has the property (\(\sigma\)-\(A)\). The paper also contains some results about the property (\(\sigma\)-\(A)\) in generalized ordered spaces.