Almost cl-supercontinuous functions

Abstract

Abstract.Reilly and Vamanamurthy introduced the class of 'clopen maps' (≡ 'cl-supercontinuous functions'). Subsequently generalizing clopen maps, Ekici defined and studied almost clopen maps (≡ almost cl-supercontinuous functions). Continuing in the spirit of Ekici, here basic properties of almost clopen maps are studied. Behavior of separation axioms under almost clopen maps is elaborated. The interrelations between direct and inverse transfer of topological properties under almost clopen maps are investigated. The results obtained in the process generalize, improve and strengthen several known results in literature including those of Ekici, Singh, and others.

Between continuity and set connectedness

“…On the other hand a condition strictly weaker than continuity is sufficient to meet the demand of a particular situation. Several strong variants of continuity occur in the lore of mathematical literature and applications of mathematics (see for example [6,12,13,14,19,20,24,27,30,32,34,38]), while others are weaker than continuity (see for example [7, 11, 15, 23, 25, 28, 35, 36, 37]), and yet others are independent of continuity (see for example [17,18,21,22,26,31]). In this paper we restrict ourselves to the study of weak variants of continuity and introduce two new weak variants of continuity called 'R-continuity' and 'F-continuity' and study their basic properties.…”

On almost cl-supercontinuous functions

“…Over the years many generalizations and variants of continuous functions between topological spaces have been introduced. Very recently, Kohli and Singh [7], have considered the class of "almost cl-supercontinuous " functions. This is a new name for the class of "almost clopen " functions introduced by Ekici [4] as a generalization of the class of "clopen continuous " mappings defined by Reilly and Vamanamurthy [11], and studied in some detail by Singh [13], under the name of cl-supercontinuous functions.…”

Rδ-Supercontinuous Functions

“…Sufficiency is immediate in view of Theorem 4.4. To prove necessity, suppose that g˝f is R δ -supercontinuous and let A subset Y of a space X is said to be δ-embedded ( [13,14] (c) Let F be any closed subset of Y . Then f´1pF q " Ť n i"1 f´1 i pF q.…”

“…(ii) δT 1 -space [13]˚if for every pair of distinct points x and y in X, there exist regular open sets U and V in X such that x P U , y R U and y P V , x R V . (iii) δT 0 -space ( [13,14]) if for every pair of distinct points x and y in X, there exists a regular open set containing one of the points x and y but not both. [3] ; if x, y P X, x R tyu, then x and y are contained in disjoint open sets.…”