Properties of prealmost \(\alpha\)-continuous and presemi-weakly continuous functions (Q2494430)

Let \((X,\tau)\) be a topological space (no separation axioms are assumed). A subset \(S\) of \((X,\tau)\) is said to be \(\alpha\)-open (respectively, semi-open) in \((X,\tau)\) if \(S\subset int (\text{cl}(\int(S)))\) (respectively, \(S \subset \text{cl}(\int(S))\)) see \textit{O. Njlstad} [Pac. J. Math. 15, 961--970 (1965; Zbl 0137.41903)], and \textit{N.Levine} [Am. Math. Monthly 7, 36--41 (1963; Zbl 0113.16304)]. \(\alpha\)-closed sets and semi-closed sets are defined in the natural way. A function \(f\) from a topological space \((X,\tau)\) into a topological space \((Y,\sigma)\) is called prealmost \(\alpha\)-continuous (respectively, presemi-weakly continuous) if for each \(x\in X\) and each \(\sigma\)-open subset \(V\) of \(Y\) containing \(f(x)\) there exists an \(\alpha\)-open (respectively, a semi-open) subset \(U\) of \((X,\tau)\) containing \(x\) such that \(f(U)\subset \text{cl} (V)\). In the paper under review some properties of these classes of functions are established in the flavor of the ones obtained in \textit{T.Noiri} [Proc. Am. Math. Soc. 46, 120-124 (1974; Zbl 0294.54013)] and [Kyungpook Math. J. 25, 123--126 (1985; Zbl 0594.54009)], and \textit{Z. Duszyński} [Kyungpook Math. J. 44, No. 2, 249--260 (2004; Zbl 1060.54505)] concerning weakly continuous, semi-weakly continuous and almost \(\alpha\)-continuous functions, respectively. For instance, among other things the author shows the following: (1) If \(f\colon (X,\tau)\rightarrow (Y,\sigma)\) is an arbitrary prealmost \(\alpha\)-continuous (respectively, presemi-weakly continuous) function, then so is the graph mapping of \(f\), (2) the Cartesian product of two prealmost \(\alpha\)-continuous (respectively, presemi-weakly continuous) functions is also a prealmost \(\alpha\)-continuous (respectively, a presemi-weakly continuous) function, and (3) if \(f\) is a prealmost \(\alpha\)-continuous (respectively, a presemi-weakly continuous) function, then the graph of \(f\) is an \(\alpha\)-closed (respectively, a semi-closed) subset of the product space \(X\times Y\).