Spaces with a \(\sigma\)-point-discrete weak base (Q946951)

The authors study spaces with \(\sigma\)-point-discrete weak bases. Let \(X\) be a topological space. For every \(x\in X\) let \({\mathcal T}_x\) be a family of subsets of \(X\) containing \(x\). If the collection \(\{{\mathcal T}_x:x\in X\}\) satisfies (1) for every \(x\in X\) the intersections of finitely many members of \({\mathcal T}_x\) belong to \({\mathcal T}_x\) and (2) \(U\subset X\) is open in \(X\) if and only if \(x\in U\) implies \(x\in T\subset U\) for some \(T\in{\mathcal T}_x\), then it is called a \textit{weak base} for \(X\). Let \({\mathcal B}=\{B_{\alpha}:\alpha\in I\}\) be a family of subsets of a space \(X\). \(\mathcal B\) is called \textit{point-discrete} if \(\{x_{\alpha}:\alpha\in I\}\) is closed discrete in \(X\) whenever \(x_{\alpha}\in{\mathcal B}_{\alpha}\) for each \(\alpha\in I\). If any compact subset of \(X\) meets at most finitely many member of \(\mathcal B\), then \(\mathcal B\) is called \textit{compact-finite}. The authors consider under what conditions a space with a \(\sigma\)-point-discrete weak base has a \(\sigma\)-compact-finite weak base, and discuss some mapping properties of spaces with \(\sigma\)-point-discrete weak bases. There are many metrization theorems for spaces with different types of bases. In this paper, the authors give some new metrization theorems. For example, they show that a space \(X\) is metrizable if and only if \(X^{\omega}\) has a \(\sigma\)-point-discrete weak base. On the other hand, the authors give an example of a non-metrizable space \(X\) such that \(X^n\) has a \(\sigma\)-point-discrete base for each \(n\in{\mathbb N}\). Finally, they give an application for paratopological groups and pose some open questions.