The open-point and bi-point-open topologies on \(C(X)\): submetrizability and cardinal functions
Denote by \(C_h(X)\) (resp. \(C_{ph}(X)\)) the \textit{open-point } (resp. \textit{bi-point-open}) topology on the space \(C(X)\) of all real-valued continuous functions on a Tychonoff space \(X\), which have subbase elements of the form \[ \{ f\in C(X): f^{-1}(r)\cap U\neq\emptyset\}, \] where \(U\) is \(X\)-open, and \(r\in \mathbb R\) (resp. \[ \{ f\in C(X): f^{-1}(r)\cap U\neq\emptyset \;\text{and} \;f(x)\in V\}, \] where \(U\) is \(X\)-open, \(x\in X\), \(V\) is \(\mathbb R\)-open, and \(r\in \mathbb R\)). The authors study submetrizability, as well as various cardinal functions, such as extent, cellularity, weight, pseudocharacter, character and tightness, for the spaces \(C_h(X),C_{ph}(X)\).
scientific article; zbMATH DE number 6509446
scientific article