On almost precontinuous functions (Q1587328)

Let \((X,\tau)\) be a topological space. A subset \(A\) of \(X\) is said to be pre-open [\textit{A. S. Mashhour}, \textit{M. E. Abd El-Monsef} and \textit{S. N. El-Deep}, Proc. Math. Phys. Soc. Egypt 53, 47-53 (1982; Zbl 0571.54011)] if \(A\subset\text{Int(Cl}(A))\). In [Acta Math. Hung. 74, No. 3, 211-219 (1997; Zbl 0924.54017)] \textit{A. A. Nasef} and \textit{T. Noiri} introduced the class of almost precontinuous functions in the sense of \textit{M. K. Singal} and \textit{A. R. Singal} [Yokohama Math. J. 16, 63-73 (1968; Zbl 0191.20802)]. A function \(f:(X,\tau)\to(Y,\sigma)\) is said to be precontinuous (resp. almost precontinuous) at \(x\in X\) if for each open (resp. regular open set) \(V\) of \(Y\) containing \(f(x)\), there exists a pre-open set \(U\) of \(X\) containing \(x\) such that \(f(U)\subset V\) (resp. \(f(U)\subset \text{Int (Cl}(V))\)). If \(f\) is precontinuous (resp. almost continuous) at every point \(x\) of \(X\) then it is called precontinuous (resp. almost precontinuous). A function \(f:X\to Y\) is almost precontinuous if \(f:X\to Y_s\) is precontinuous, where \(Y_s\) denotes the semi-regularization of \(Y\). In this paper, the authors investigate some properties of almost precontinuous functions.