On \(X\)-\(\vartheta\)-splitting and \(X\)-\(\vartheta\)-jointly continuous topologies on function spaces (Q2785363)

Let \(Y,Z\) be topological spaces and let \(f\) be a map of \(Y\) into \(Z\). Then \(f\) is \(\vartheta\)-continuous at \(y\in Y\) if for every open neighbourhood \(V\) of \(f(y)\) there exists an open neighbourhood \(U\) of \(y\) such that \(f(\text{Cl} (U))\subseteq \text{Cl}(V)\) (by \(\text{Cl}(A)\) is denoted the closure of \(A\) in the topological space \(Y)\). The map \(f\) is \(\vartheta\)-continuous on \(Y\) if it is \(\vartheta\)-continuous at each point of \(Y\). (See for example [\textit{S. Fomin}, Ann. Math., II. Ser. 44, 471-480 (1943; Zbl 0061.39601); \textit{S. Iliadis} and \textit{S. Fomin}, Russ. Math. Surv. 21, No. 4, 37-62 (1966); translation from Usp. Mat. Nauk 21, No. 4(130), 47-76 (1966; Zbl 0171.21304); \textit{J. E. Joseph}, Math. Chron. 8, 99-117 (1979; Zbl 0441.54001)]. By \(f\mathbb{S}\) is denoted the Sierpiński space, that is, the set \(\{0,1\}\) equipped with the topology \(\tau(\mathbb{S}) \equiv\{\emptyset, \{0,1\}, \{1\}\}\), and by \(\mathbb{D}\) the set \(\{0,1\}\) with the trivial topology. NEWLINENEWLINENEWLINEIn the paper a relation on the set \(\Theta(Y,Z)\) of all \(\vartheta\)-continuous functions of a topological space \(Y\) into a topological space \(Z\) is defined and the connection of this relation with the notions of \(X\)-\(\vartheta\)-splitting and \(X\)-\(\vartheta\)-jointly continuous topologies on this set is studied, where \(X=\mathbb{S}\) or \(X=\mathbb{D}\).