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On \(\gamma\)-normal spaces. (English) Zbl 1150.54023

Various notions of normality of a topological space can be defined by requiring that disjoint closed sets can be separated by certain generalized open sets. In the present paper the author considers \(\gamma\)-open sets (also known as \(b\)-open sets) and investigates \(\gamma\)-normal spaces where disjoint closed sets can be separted by disjoint \(\gamma\)-open sets. In the first part the author observes that both \(p\)-normal spaces and \(s\)-normal spaces are \(\gamma\)-normal spaces but not conversely, and also provides a standard characterization of \(\gamma\)-normal spaces. It would have been very helpful if the author also provided examples for such spaces having more than 3 or 4 points in order to illustrate that this class of spaces indeed has some relevance.
A large part of this paper is devoted to consider various functions with respect to \(\gamma\)-normal spaces. Also, certain other types of sets such as \(g\gamma\)-closed sets and \(\gamma g\)-closed sets come into play. All these notions are illustrated by several examples, and again, here it would be useful to know whether these notions also have applications in spaces having infinitely many points.

MSC:

54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54A05 Topological spaces and generalizations (closure spaces, etc.)
54C08 Weak and generalized continuity
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