On function spaces topologies in the setting of Čech closure spaces (Q554396)

A set \(X\) is called a Čech closure space \((X, u)\) in case there is an operator \(u\) defined on all subsets \(A\subset X\) with \(u(A)\subset X\) such that {\parindent7.5mm \begin{itemize}\item[(i)] \(u(\emptyset)= \emptyset\); \item[(ii)] \(u(A)\supset A\); \item[(iii)] \(u(A\cup B)= u(A)\cup u(B)\). \end{itemize}} It is not required that \(u(u(A))= u(A)\). Sets of the form \(X\setminus u(A)\) are said to be open in \(X\). Let \((Y, v)\) be another closure space and \({\mathcal F}\) a collection of functions \(f\) from \((X, u)\) to \((Y, v)\). Separation axioms on \(X\) from \(T_0\) up to normal are defined in terms of \(u\), and those for \({\mathcal F}\) are defined in terms of the product closure operator on \(X\times Y\). Among the topologies on \({\mathcal F}\) studied in the present paper is the graph-open topology with basis \(\{f\in{\mathcal F}: G(f)\subset W\}\), where \(G(f)\) is the graph of \(f\) in \(X\times Y\) and \(W\) is open in \(X\times Y\). Various relations between separation properties in \(X\), \(Y\), and \({\mathcal F}\) are obtained. We say that \(f\) is continuous in case \(f(u(A))\subset v(f(A))\) for every \(A\subset X\). We denote by \({\mathcal C}(X, Y)\) the set of all continuous functions from \(X\) to \(Y\). The authors show, among other things, that if \((X, u)\) is regular, \(Y\) is a topological space, and \({\mathcal F}\subset{\mathcal C}(X, Y)\), then the compact-open topology on \({\mathcal F}\) is contained in the graph-open topology.