Some cardinal invariants on the space \(C_{\alpha}(X,Y)\) (Q1779526)

Let \(C_\alpha(X,Y)\) be the set of all continuous functions from \(X\) to~\(Y\) endowed with the set-open topology where \(\alpha\)~is a~compact network closed on closed subsets and finite unions (in particular, if \(\alpha\) consists of all compact sets, the topology of \(C_\alpha(X,Y)\) is the usual compact-open topology). The authors introduce two properties of triples \((\alpha,X,Y)\) which enable to obtain some equalities and inequalities between some topological cardinal invariants of \(C_\alpha(X,Y)\). As a~consequence they obtain several characterizations of the condition ``\(C_\alpha(X,Y)\) is a~first countable space.''