## 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)

### MSC:

 54C08 Weak and generalized continuity

### Citations:

Zbl 0571.54011; Zbl 0524.54016; Zbl 0576.54014