cl-Supercontinuous Functions

Abstract

Abstract. Basic properties of cl-supercontinuity, a strong variant of continuity, due to Reilly and Vamanamurthy [Indian J. Pure Appl. Math., 14 (1983), 767-772], who call such maps clopen continuous, are studied. Sufficient conditions on domain or range for a continuous function to be cl-supercontinuous are observed. Direct and inverse transfer of certain topological properties under cl-supercontinuous functions are studied and existence or nonexistence of certain cl-supercontinuous function with specified domain or range is outlined.2000 AMS Classification: 54C08, 54C10, 54D10, 54D20, 54D30.

R-spaces and closedness/completeness of certain function spaces in the topology of uniform convergence

“…Furthermore, Kohli and Aggarwal in [14] proved that the function space SC(X, Y ) of quasicontinuous ( ≡ semicontinuous) functions, C α (X, Y ) of α-continuous functions, and L(X, Y) of cl-supercontinuous functions are closed in Y X in the topology of uniform convergence. In this section we strengthen the results of [14] and show that the set [32] if for each x ∈ A there exists a clopen set H such that…”

“…Let X be a topological space. Any intersection of clopen sets in X is called cl-closed [32]. An open subset U of X is said to be r cl -open [37] if for each x ∈ U there exists a cl-closed set H containing x such that H ⊂ U ; equivalently U is expressible as a union of cl-closed sets.…”

“…(ii) ultra Hausdorff [35] if every pair of distinct points in X are contained in disjoint clopen sets. (iii) R z -space ( [20,33]) if for each open set U in X and each x ∈ U there exists a z-closed set H containing x such that H ⊂ U ; equivalently U is expressible as a union of z-closed sets. (iv) R δ -space [19] if for each open set U in X and each x ∈ U there exists a δ-closed set H containing x such that H ⊂ U ; equivalently U is expressible as a union of δ-closed sets.…”

“…We reflect upon interrelations and interconnections that exist among R cl -spaces and separation axioms which already exist in the lore of mathematical literature and lie between zero dimensionality and R 0 -spaces. The class of R cl -spaces properly contains each of the classes of zero dimensional spaces and ultra Hausdorff spaces [35] and is strictly contained in the class of R 0 -spaces ( [20,33]) which in its turn properly contains each of the classes of functionally regular spaces ( [3,39]) and functionally Hausdorff spaces. The organization of the paper is as follows: Section 2 is devoted to preliminaries and basic definitions.…”

Generalizations of Z-supercontinuous functions and Dδ-supercontinuous functions

“…In this paper we introduce two new variants of continuity which represent generalizations of the notions of z-supercontinuity and D δ -supercontinuity and are independent of continuity and coincide with zsupercontinuity and D δ -supercontinuity, respectively if the range is a semiregular space. The class of almost z-supercontinuous functions besides containing the class of z-supercontinuos functions contains the class of almost clopen (≡ almost cl-supercontinuous [34]) functions defined by Ekici [4]. Characterizations and basic properties of almost z-supercontinuous (almost D δ -supercontinuous) functions are elaborated in Section 3 and their place in the hierarchy of variants of continuity is discussed.…”