A topology generated by selections
In this paper the authors continue their investigations from the paper [Appl. Gen. Topol. 2, 205-218 (2001; Zbl 0993.54019)]. Let \((X, {\mathcal T})\) be a \(T_1\) topological space. Let \({\mathcal F}_2 (X)\) be the family of at most two-point subsets of \(X\) equipped with the Vietoris topology and let \(f: {\mathcal F}_2 (X) \rightarrow X\) be a selection. The selection \(f\) defines an order-like relation \(\preceq _f\) on \(X\) by \(x \preceq _f y\) if and only if \(f(\{x,y\})=x\). The ``open'' intervals \((x, \rightarrow )=\{y \in X: x \preceq _f y \) and \(y \neq x\}\) and \(( \leftarrow ,x)=\{y \in X: y \preceq _f x \) and \(y \neq x \}\) generate the topology \({\mathcal T}_f \). Properties of this topology are investigated. A number of interesting examples is presented, for instance, continuous selections \(f\) on various spaces \((X, {\mathcal T})\) such that \({\mathcal T}_f \neq {\mathcal T}\).
scientific article; zbMATH DE number 5013562