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Semi-generalized continuous maps in topological spaces. (English) Zbl 0883.54015

N. Levine [Rend. Circ. Mat. Palermo, II. Ser. 19, 89-96 (1970; Zbl 0231.54001)] has defined a subset \(A\) to be \(g\)-closed if \(\text{Cl} (A)\subset O\) when \(A\subset O\) and \(O\) is open. The complement of a \(g\)-closed set is called \(g\)-open. The purpose of this paper is to introduce and study the concepts of two new classes of maps, namely the class of \(sg\)-continuous maps, which includes the class of continuous maps, and the class of \(sg\)-irresolute maps defined analogously to irresolute maps. Moreover, we introduce the concepts of \(sg\)-compactness and \(sg\)-connectedness of topological spaces. Among the theorems proved are the following:
(A) The following are equivalent: (i) \(X\) is \(sg\)-connected; (ii) \(X\) and \(\emptyset\) are the only subsets of \(X\) which are both \(sg\)-open and \(sg\)-closed; (iii) each \(sg\)-continuous map of \(X\) into a discrete space \(Y\) with at least two points is a constant map.
(B) \(sg\)-connectedness is preserved under \(sg\)-irresolute surjections.

MSC:

54C08 Weak and generalized continuity
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)

Citations:

Zbl 0231.54001
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