\((\delta, \delta ')\)-continuity on generalized topological spaces
W. K. Min studies continuity for generalised topological spaces where the empty set is open and the arbitrary union of open sets is again open; intersections of open sets need not to be open. The resulting interior-operator \(i(A)\) being the union of all open subsets of \(A\) satisfies that \(i(A) \subseteq A\), \(A \subseteq B \Rightarrow i(A) \subseteq i(B)\) and \(i(i(A)) = A\). Hence the closure-operator in such a generalised topology is just any closure operator satisfying the usual axioms and one can therefore identify the generalised topological spaces with the spaces of such closure operators. Given an open topology \(\mu\) of this type, W. K. Min derives another topology \(\delta\) which contains all those subsets \(A\) of \(X\) such that for each \(x \in A\) there is a \(\mu\)-closed set \(B \subseteq A\) with \(x\) being in the \(\mu\)-interior of \(B\). Given now two generalised topological spaces \((X,\mu)\) and \((Y,\mu')\), the author studies when a function \(f: X \rightarrow Y\) is \((\delta,\delta')\)-continuous with respect to \(\delta,\delta'\) derived from \(\mu,\mu'\). The notion of \((\delta,\delta')\)-continuity is independent of \((\mu,\mu')\)-continuity, but both types of continuities imply that the function is almost \((\mu,\mu')\)-continuous and weakly \((\mu,\mu')\)-continuous, where these notions were introduced by the author in prior work. The notions of \((\mu,\mu')\)-continuity, \((\delta,\delta')\)-continuity and almost \((\mu,\mu')\)-continuity coincide on a suitable generalisation of the notion of regular spaces to this form of generalised topology.
scientific article; zbMATH DE number 5857801