## 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.)

### Keywords:

$$sg$$-continuous maps; $$sg$$-irresolute maps

Zbl 0231.54001
Full Text: