\textit{cs}-Regular families, \textit{cs}-finite families and the images of metric spaces (Q2217224)

A topological space \(Y\) is \(cs\)-metrizable if there are a metric space \(X\), a continuous mapping \(X \xrightarrow{f} Y\), and a mapping \(Y \xrightarrow{s} X\) such that the composition \[ Y \xrightarrow{s} X \xrightarrow{f} Y \] is the identity mapping and, for every convergent sequence \((y_n)_{n \in \mathbb{N}}\) in \(Y\), the sequence \((s(y_n))_{n \in \mathbb{N}}\) is convergent in \(X\). The authors find characterizing properties of \(cs\)-metrizable spaces. To formulate one of the typical results of the paper we need some definitions. Let \((X, \tau)\) be a topological space. A set \(A \subseteq X\) is called sequentially open in \(X\) if, for every \(x \in A\), each sequence converging to \(x\) is eventually in \(A\). A family \(\mathcal{P}\) of subsets of \(X\) is called \(so\)-regular if, for each pair \(x\), \(U\) with \(x \in U \in \tau\), there exists a sequentially open \(V \ni x\) for which the family \[ \{P \in \mathcal{P} \colon P \nsubseteq U \text{ and } V \cap P \neq \varnothing\} \] is finite. If a family \(\mathcal{F}\) is a network for \((X, \tau)\) and each element of \(\mathcal{F}\) is sequentially open, then \(\mathcal{F}\) is said to be an \(so\)-network. \textbf{Theorem.} A topological space is \(cs\)-metrizable if and only if this space has an \(so\)-regular \(so\)-network. To obtain future distinct characterizations of \(cs\)-metrizable spaces, the authors consider some other classes of networks and related mappings. The paper contains also an interesting short survey on the history of the topological theory of generalized metric spaces.