Feeble and strong forms of preirresolute functions. (English) Zbl 0885.54010

A subset \(A\) of a topological space \(X\) is said to be pre-open if \(A\subset \text{Int(Cl}(A))\) [A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, Proc. Math. Phys. Soc. Egypt 53, 47-53 (1982; Zbl 0571.54011)]. \(A\) is preclosed if \(X-A\) is pre-open and \(\text{Pcl}(A)= \bigcap \{B: B\) preclosed in \(X\) and \(B\supset A\}\) [N. El-Deeb, I. A. Hasanein, A. S. Mashhour and T. Noiri, Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 27(75), 311-315 (1983; Zbl 0524.54016)]. A function \(f:X\to Y\) is said to be pre-irresolute [I. L. Reilly and M. K. Vamanamurthy, Acta Math. Hung. 45, 27-32 (1985; Zbl 0576.54014)] if \(f^{-1}(V)\) is pre-open for every pre-open set \(V\) of \(Y\). In this paper two forms of pre-irresolute functions are introduced.
Definition: A function \(f:X\to Y\) is said to be quasi-pre-irresolute (strongly pre-irresolute) at \(x\in X\) if for each pre-open set \(V\) containing \(f(x)\) there exists a pre-open set \(U\) of \(X\) containing \(x\) such that \(f(U)\subset \text{Pcl}(V)\) \((\text{Pcl}(U)\subset V)\).
Some characterizations and their basic properties are obtained. The intrinsic connection with other weakened forms of continuity (precontinuity, quasi-precontinuity or almost weak continuity, pre-irresoluteness and \(\alpha\)-irresoluteness) are also studied.
Reviewer: V.Popa (Bacau)


54C08 Weak and generalized continuity