F-supercontinuous functions

Abstract

A strong variant of continuity called 'F -supercontinuity' is introduced. The class of F -supercontinuous functions strictly contains the class of z-supercontinuous functions (Indian J. Pure Appl. Math. 33 (7) (2002), 1097-1108) which in turn properly contains the class of cl-supercontinuous functions (≡ clopen maps) (Appl. Gen. Topology 8 (2) (2007), 293-300; Indian J. Pure Appl. Math. 14 (6) (1983), 762-772). Further, the class of F -supercontinuous functions is properly contained in the class of R-supercontinuous functions which in turn is strictly contained in the class of continuous functions. Basic properties of F -supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity, which already exist in the mathematical literature, is elaborated. If either domain or range is a functionally regular space (

Rδ-Supercontinuous Functions

“…The function f is said to be R δ -supercontinuous, if it is R δ -supercontinuous at each x P X. However, none of the above implications in general is reversible, which is either well known (see [11,16,17,29]) or follows from the following observations and examples.…”

“…An open set U of a space X is said to be F -open [16] (r-open [17]) if for each x P U there exists a zero(closed) set Z in X such that x P Z Ă U , equivalently if U is expressible as a union of zero (closed) sets in X. A subset A of a space X is said to be regular open if it is the interior of its closure, i.e.…”

“…r δ -quotient topology and r δ -quotient spaces Several variants of quotient topology occur in the lore of mathematical literature (see [16,17,20]). In this section, we introduce a new variant of quotient topology called 'r δ -quotient topology' which strictly lies between r z -quotient topology [30] and r-quotient topology [17].…”

Between continuity and set connectedness

“…An open subset U of X is said to be r-open [20] if for each x ∈ U there exists a closed set B such that x ∈ B ⊂ U or equivalently, if U is expressible as a union of closed sets. An open subset W of X is said to be F-open [19] if for each x ∈ W there exists a zero set Z such that x ∈ Z ⊂ W or equivalently, if W is expressible as a union of zero sets.…”

“…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.…”