DEMONSTRATIO MATHEMATICA Vol. XLIII No 3 2010 J. K. Kohli, D. Singh, J. Aggarwal R-SUPERCONTINUOUS FUNCTIONS Abstract. A new strong variant of continuity called 'i?-supercontinuity' is intro- duced. Basic properties of R-supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity that already exist in the literature is elaborated. It is shown that fl-supercontinuity is preserved under the restriction, shrinking and expan- sion of range, composition of functions, products and the passage to graph function. The class of .R-supercontinuous functions properly contains each of the classes of (i) strongly 0-continuous functions introduced by Noiri and also studied by Long and Herrington; (ii) D-supercontinuous functions; and (iii) F-supercontinuous functions; and so include all z-supercontinuous functions and hence all clopen maps (= cl-supercontinuous func- tions) introduced by Reilly and Vamnamurthy, perfectly continuous functions defined by Noiri and strongly continuous functions due to Levine. Moreover, the notion of r-quotient topology is introduced and its interrelations with the usual quotient topology and other variants of quotient topology in the literature are discussed. Retopologization of the do- main of a function satisfying a strong variant of continuity is considered and interrelations among various coarser topologies so obtained are observed. 1. Introduction Strong variants of continuity arise in diverse situations in mathemat- ics and applications of mathematics and are interspersed throughout the literature. In many situations in topology, analysis and in many other branches of mathematics and its applications, continuity is not enough and a strong form of continuity is required to meet the demand of a particular situation. Hence it is of considerable significance both from applications viewpoint and intrinsic considerations to formulate and study new strong variants of continuity. In this paper, we introduce one such strong variant of continuity called 'R-supercontinuity' and study the basic properties of .R-supercontinuous functions and elaborate on their place in the hierarchy Key words and phrases: strongly 0-continuous function, supercontinuous function, Z)-supercontinuous function, F-supercontinuous function, iio-space, r-quotient topology, ¿-quotient topology, 0-quotient topology, r-open set. 2000 Mathematics Subject Classification: 54C08, 54C10, 54D10, 54D20. This research was partially supported by University Grants Commision, India. 704 J. K. Kohli, D. Singh, J. Aggarwal of strong variants of continuity that already exist in the literature. The class of ii-supercontinuous functions properly includes each of the classes of .D-supercontinuous functions [15], strongly ¿^-continuous functions intro- duced by Noiri [28] and F-supercontinuous functions [18]. Thus, in turn the class of ii-supercontinuous functions contain, all z-supercontinuous func- tions [14], Dj-supercontinuous functions [16], cl-supercontinuous functions [33] (= clopen maps [30]) initiated by Reilly and Vamanamurthy, perfectly continuous functions defined by Noiri [29] and strongly continuous functions due to Levine [22], Section 2 is devoted to basic definitions and preliminaries. In Section 3, we introduce the notion of 'ii-supercontinuous function' and elaborate on its place in the hierarchy of strong variants of continuity that already exist in the mathematical literature. We study basic properties of ii-supercontinuous functions in Section 4. It is shown that (i) ii-supercontinuity is invari- ant under restrictions, shrinking and expansion of range and composition of functions; (ii) a function into a product space is ii-supercontinuous if and only if its composition with each projection map is ii-supercontinuous; and (iii) the graph function g of / is ii-supercontinuous if and only if / is ii-supercontinuous and X is an iio-space. The interplay between ii- supercontinuity and topological properties is considered in Section 5. In Section 6, properties of graph of an ii-supercontinuous function are dis- cussed. The notion of r-quotient topology is introduced in Section 7 and its interrelations with the standard quotient topology and other variants of quotient topology in the literature are observed. In Section 8, we consider the retopologization of the domain of an ii-supercontinuous function (or sat- isfying some other strong form of continuity) in such a way that it is simply a continuous function. We discuss the interrelations and interplay among various coarser topologies obtained in this way and substantiate with exam- ples to show that all these coarser topologies in general are distinct. We conclude the section with alternative proofs of certain results of preceding sections. Finally an appendix is added in which we include the basics for the category UPS (= the category whose objects are complete lattices equipped with Scott topologies and morphisms are Scott continuous maps). 2. Basic definitions and preliminaries A collection fi of subsets of space X is called an open complementary system [11] if (5 consists of open sets such that for every B £ ¡3, there exist BI,B2,...£/3 with B = : i e N}. A subset U of a space X is called strongly open Fa-set [11] if there exists a countable open complimentary system (3(U) with U € (3(U). The complement of a strongly open Fa-set is referred to as a strongly closed G$-set. A subset H of a space X is called a R-supercontinuous functions 705 regular Gg-set [26] if H is an intersection of a sequence of closed sets whose = interiors contains H, i.e. H = ^n fl^Li where each Fn is a closed subset of X. The complement of a regular G^-set is called a regular Fa-set. An open set U of a space X is said to be F-open [18] if for each x G X there exists a zero set Z in X such that x G Z C U, equivalently if U is expressible as a union of zero sets in X. A subset A of a space X is said to be regularly open if it is the interior of its closure, i.e., A = A . A point x £ X is called a 6-adherent point [36] of a set A C X if every closed neighbourhood of x intersects A. Let cle A denote the set of all ^-adherent points of A. The set A is called 9-closed if A = clg A. The complement of a 0-closed set is referred to as 6-open. It is easily verified that a set is 0-open if and only if it contains a closed neighbourhood of each of its points. A topological space X is said to be an Alexandroff space [1] if arbitrary intersection of open sets in X is open. Alexandroff spaces are referred to as saturated spaces in [25]. Several weak and strong variants of continuity occur in the literature. Here we include definitions of only those strong variants of continuity which will be dealt with in this paper. DEFINITIONS 2.1. A function f : X —>Y from a topological space X into a topological space Y is said to be (a) strongly continuous [22] if f(A) C f(A) for each subset A of X, (b) perfectly continuous [29] if f~l(V) is clopen in X for every open set FCF, (c) cl-supercontinuous [33] (= clopen continuous [30]) if for each x G X and each open set V containing f(x) there is a clopen set U containing x such that /([/) C V, (d) z-supercontinuous [14] if for each x G X and for each open set V contain- ing f(x), there exists a cozero set U containing x such that f(U) C V, (e) Dg-supercontinuous [16] if for each x G X and for each open set V containing f(x), there exists a regular F^-set U containing x such that f(U) C V, (f) D-supercontinuous [15] if for each x G X and each open set V containing f(x) there exists an open i^-set V containing x such that f(U) C V, (g) D*-supercontinuous [32] if for each x G X and each open set V contain- ing f(x) there exists a strongly open FCT-set V containing x such that f(u) c v, (h) strongly 0-continuous [28] if for each x G X and for each open set V con- taining f(x), there exists an open set U containing x such that f(U) C V, (i) F-supercontinuous [18] if for each x G X and each open set V containing f(x) there exists an F-open set U containing x such that f(U) C V, 706 J. K. Kohli, D. Singh, J. Aggarwal (j) supercontinuous [27] if for each x € X and for each open set V containing f(x), there exists a regular open set U containing x such that f(U) C V. DEFINITIONS 2.2. A topological space X is said to be (1) an Ro-space ([7], [31]) if for each open set U in X, x € U implies {x} C U, (2) an Ri-space [7] if x, y € X, x £ {y}, then x and y are contained in disjoint open sets, (3) a functionally regular space ([2], [4]) if for each closed set A in X and a point x $ A there exists a continuous real valued function / defined on X such that f(x) £ f(A)\ equivalently for each x e X and for each open set U containing x there exists a zero set Z such that x € Z C U, (4) D-regular space [11] if it has a base of open Fa- sets, (5) functionally Hausdorff if for x, y G X, x ^ y there exists a continuous function / : X —> [0,1] such that f(x) ^ f(y), (6) semiregular if it has a base of regular open sets.