On \(\theta\)-precontinuous functions (Q1607742)

A subset of a topological space \((X,\tau)\) is said to be preopen if \(A\subset \text{Int}(\text{Cl}(A))\). A function \(f:(X,\tau)\to (Y,\sigma)\) is said to be precontinuous [\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)] (resp. almost precontinuous [\textit{A. A. Nasef} and \textit{T. Noiri}, Acta Math. Hung. 74, No. 3, 211-219 (1997; Zbl 0924.54017)], weakly precontinuous [\textit{V. Popa} and \textit{T. Noiri}, Demonstr. Math. 25, No. 1/2, 241-251 (1992; Zbl 0789.54014)] or quasi-precontinuous [\textit{M. C. Pal} and \textit{P. Bhattacharyya}, Bull. Malays. Math. Soc., II. Ser. 19, No. 2, 63-75 (1996; Zbl 0885.54010)]) if for each \(x\in X\) and each open set \(V\) containing \(f(x)\), there exists a preopen set \(U\) containing \(x\) such that \(f(U)\subset V\) (resp. \(f(U)\subset \text{Int}(\text{Cl}(V))\), \(f(U)\subset\text{Cl}(V)\)). In this paper the author introduces the notion of a \(\theta\)-precontinuous function. A function \(f: (X,\tau)\to (Y,\sigma)\) is \(\theta\)-precontinuous if for each \(x\in X\) and each open set \(V\) of \(Y\) containing \(f(x)\), there exists a preopen set \(U\) of \(X\) containing \(x\) such that \(f(\text{pCl}(U))\subset \text{Cl}(V)\), where \(\text{pCl}(U)\) denotes the preclosure of \(U\). The class of \(\theta\)-precontinuous functions is contained in the class of weakly precontinuous functions and contains the class of almost precontinuous functions. Some characterizations and basic properties of these new types of functions are obtained. The relationships between these functions and several forms of precontinuity are investigated.