On some questions concerning preopen sets.(English)Zbl 0718.54004

A subset S of a topological space (X,$$\tau$$) is called preopen if $$S\subseteq int(cl S)$$. In this paper four various questions, K1-K4, about preopen sets raised by Katětov are investigated. For questions K2-K4 partial answers are given, the question K1 “Find necessary and sufficient conditions under which every preopen set is open” is answered completely in the following way (Theorem 1 and 4): “For a topological space (X,$$\tau$$) the following statements are equivalent: (a) Every preopen set is open. (b) Every dense set is open. (c) Every continuous function $$f:(Z,z)\to (X,\tau)$$ is strongly precontinuous, where (Z,z) is any topological space.” - The notion of strong precontinuity was introduced by the author as follows: a mapping $$f:(Z,z)\to (X,\tau)$$ is called strongly precontinuous if the inverse image of each preopen set in X is open in Z.
Reviewer: L.Skula (Brno)

MSC:

 54A05 Topological spaces and generalizations (closure spaces, etc.) 54C10 Special maps on topological spaces (open, closed, perfect, etc.)