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.
3. /¿-supercontinuous functions An open subset A of a space X is said to be r-open if it is expressible as a union of closed sets. The complement of an r-open set will be referred to as r-closed. It is easily verified that every 0-open set is r-open and an r-open set is open. However, reverse implications need not be true as is well exhibited by the following examples. EXAMPLE 3.1. Let X denote the space of Smirnov's deleted sequence topology [35, p. 86]. Then (-e,e) \ A, where e > 0 and A = : n € N}, is an r-open set which is not #-open. EXAMPLE 3.2. Let X = {a, b} denote the two point Sierpinski space with a as an isolated point. Then {a} is an open set in X which is not r-open. Similarly if X is the real line R endowed with the right order topology [35, p. 74], then any ray (a, oo) (a € R) is an open set in X which is not r-open. Moreover, if X is an i?o-space; in particular if X is a Ti-space or a Hausdorff space, then every open set in X is r-open. DEFINITION 3.3. A function / : X —> Y from a topological space X into topological space Y is said to be R-supercontinuous at a point x £ X, if for each open set V containing /(x) there exists an r-open set U containing x such that f(U) C V. The function / is said to be R-supercontinuous, if it is R~ supercontinuous at each x 6 X. The following diagram well illustrates the interrelations that exist be- tween ii-supercontinuity and other strong variants of continuity that al- ready exist in the literature and well reflects upon the place enjoyed by R-super continuous functions 707 ii-supercontinuity in this hierarchy. The following implications are either well known or follow from definitions.
strongly continuous I perfectly continuous I clopen map (s cl-supercontinuous) 4 Ds-supercontinuous 4 z-supercontinuous i D*-supercontinuous I strongly 0-continuous D-supercontinuous F-supercontinuous I \ 1 supercontinuous R-supercontinuous
However, none of the above implications in general is reversible, which is either well known or follow from the following observations and examples.
OBSERVATIONS AND EXAMPLES 3.4. If either X or Y is an iio-space, then every continuous function / : X —> Y is /i-supercontinuous. Thus every morphism in a subcategory of TOP (= the category of topological spaces and continuous maps) which is a subcategory of the subcategory of i?o-spaces is .R-supercontinuous. On the other hand there exist subcategories of TOP in which no morphism is /¿-supercontinuous except the constant ones. One such subcategory of TOP is UPS the category whose objects are complete lattices en- dowed with Scott topology and whose morphisms are Scott-continuous maps1. 3.5. Let X = Y = R be the set of all real numbers with the right ray topology. Then X is not an iio-space. The identity function / defined on X is continuous but not .R-supercontinuous. Similarly, if X denotes a non-discrete Alexandroff space, then the identity mapping defined on X is continuous but not .R-supercontinuous. 3.6. Let X = {(x,y)\y > 0,x,y € Q} and fix some irrational number 9. The irrational slope topology [35, p. 93] r on X is generated by e-
neighbourhoods of the form Ne((x,y)) = {(x, y)}
U Be(x + y/0) U Be(x - y/0), where Be(£) = {r € Q : \r - < e},Q being the ra- tional on x-axis. Then (X, r) is a Hausdorff space which is not func- 1For the definition of Scott topology and Scott continuous maps we refer the interested reader to [10] or Appendix (Section 9). 708 J. K. Kohli, D. Singh, J. Aggarwal
tionally Hausdorff and hence not a functionally regular space. The identity mapping defined on X is i?-supercontinuous but not F-super- continuous. 3.7. Let (X, r) be the topological space formed by adding to the ordinary closed unit interval [0,1] another right end point say 1*, with the sets (a, 1) U {1*} as a local neighbourhood basis. This topology is known as Telophase topology [35, p. 92], Then X is a Ti-space and hence an iio-space. The space X is not a semiregular space because the points 1 and 1* do not have a base of regular open sets. The identity function / defined on X is ii-supercontinuous but not supercontinu- ous. 3.8. Let X = {a,b,c,d} be equipped with topology r = {, X, {a, b}, {«¿}, {a,b,d}} and let / denote the identity function defined on X. Then / is a supercontinuous function but not an ii-supercontinuous function. Thus in view of Example 3.7 supercontinuity and ii-supercontinuity are independent notions. 3.9. Let X be the product of the mountain chain space due to Helder- mann [11] and N = the set of positive integers) equipped with cofi- nite topology. The space X is a Ti-space which is neither a regular space nor a D-regular space. The identity function defined on X is an ii-supercontinuous function but neither a D-supercontinuous function nor a strongly ^-continuous function.
4. Basic properties of ii-supercontinuous functions
THEOREM 4.1. For a function f : X —>Y from a topological space X into a topological space Y, the following statements are equivalent (i) / is R-supercontinuous. (ii) f~(V) is r-open for every open set V C Y. (iii) f~1(F) is r-closed for every closed set F CY. (iv) /_1(5) is r-open for every subbasic open set S in Y. Proof. Easy. •
DEFINITION 4.2. Let X be a topological space and let A c X. A point x € X is said to be an r-adherent point of A if every r-open set containing x intersects A. Let AR denote the set of all r-adherent points of the set A. Then A C AR and a set A is r-closed if and only if A = AR. Clearly every adherent point of a set A is an r-adherent point of A and every r-adherent point of the set A is a ^-adherent point of A. However, in general the reverse implications are not true as is shown by the following examples. R-supercontinuous functions 709
EXAMPLE 4.3. Let X denotes the two point Sierpinski space considered in Example 3.2. Then the point a is an r-adherent point of the set {6} but not an adherent point. Similarly, if X denotes the real line endowed with right order topology discussed in Example 3.2, then 2 is an r-adherent point of the singleton {1} but not an adherent point. Infact in this case every point of X \ {1} is an r-adherent point of the set {1}.
EXAMPLE 4.4. Let X denote the space of Smirnov's deleted sequence topology considered in Example 3.1. Then 0 is a ^-adherent point of the set A = { ^ : n £ N} but not an r-adherent point of A.
THEOREM 4.5. For a function f : X —> Y, the following statements are equivalent. (a) / is R-supercontinuous. (b) f{AR) C f(A) for every set Ac X. l (c) (f-\B))R C f~ (B) for every set BcY. Proof. (a)=S>(b). Since f(A) is closed in Y, by Theorem 4.1 /~1(/(^)) is 1 an r-closed set in X. Again since A C f~ (f(A)),AR C = /"H/04)) and so/(¿H) C/(A). l l (b)=Kc). Let B_C Y. Then f{{f~ {B))R) c f{f~ (B)) c B and hence
(.f-\B))Rcf~\B). (c)=»(a). Let C be any closed subset of Y. Then (/-1(C))R C = f~\C). Again f-\C) C f^{C) C U~\C))R, and hence f~\C) = 1 (f~ {C))R. Thus f-\C) is r-closed and so / is .R-supercontinuous. •
DEFINITION 4.6. A filter base B is said to R-converge to a point x G X (written as B —» x) if every r-open set containing x contains a member of B.
THEOREM 4.7. A function f : X —> Y is R-supercontinuous if and only if f(B) —> f(x), for each x and for each filter base B in X that R-converges to x. Proof. Assume that / is .R-supercontinuous and let B —> x. Let V be an open set in Y containing f(x). By Theorem 4.1, f~l(V) is an r-open set containing x. Since B —> x, there exists B € B such that x € B C f~l(V) and so f(B) C V. This shows that f(B) -* f(x). Conversely, let V be an open subset of Y containing f(x). Let B be the filter generated by the filter base J\fx consisting of all r-open subsets of X containing x. By hypothesis, f(B) —> f(x) and so there exists a member f(N) of f{Mx) such that f{N) C V. Choose B 6 Mr such that B C N. Since B is an r-open set containing x and f(B) C f(N) C V,f is an R- supercontinuous function. • 710 J. K. Kohli, D. Singh, J. Aggarwal
THEOREM 4.8. If f : X —> Y is R-supercontinuous and g : Y —> Z is a continuous function, then the composition g o f is R-supercontinuous. In particular, composition of two R-supercontinuous functions is R-super- continuous. Proof. Let W be an open set in Z. Then g~x(W) is an open set in Y. By .R-supercontinuity of /, /-1(<7-1(W)) = (g o is r-open in X and so g o / is .R-supercontinuous. • REMARK 4.9. However, i?-supercontinuity of g o / need not imply even continuity of /. For, let, X = Y = Z = R, where R denotes the set of all real numbers. Let X,Y and Z be endowed with cocompact topology r, usual topology u and cofinite topology r*, respectively. Let / : (X, r) —>• (F, r) and g : (Y,u) —> (Z,r*) denote the identity functions. Then go f and g are .R-supercontinuous but / is not even a continuous function. DEFINITION 4.10. A function / : X —> Y is said to be (1) R-open (R-closed) if f(A) is open (closed) in Y for every r-open (r- closed) set A in X. (2) 0-open if f(A) is open in Y for every 0-open set A in X. Every open function is .R-open and every .R-open function is 0-open. However, reverse implications are not true as is well exhibited by the follow- ing examples. EXAMPLE 4.11. Let X = {a,b,c} be endowed with topology r = {(f), X, {a, b}, {6, c}, {c}, {b}} and let Y = {x, y, z} be equipped with topology r* = {4, Y, {x, y}, {z}}. Let / : (X, r) -> (Y, T*) be defined by f(a) = x, f(b) = y, f(x) = z. Then / is an .R-open function but not an open function. EXAMPLE 4.12. Let X denote the space of Smirnov's deleted sequence topology considered in Example 3.1. Let Y = {0,1,2} be endowed with the topology r = {0, Y, {1}, {0,1}}. Define a function / : X —»• Y by [l, if x ^ U {0}), f(x) = i 0, if X e A, (2, if x = 0. It is easily verified that / is a 0-open function but not an .R-open function. THEOREM 4.13. Let f : X —» Y be an R-open, R-supercontinuous surjec- tion and let g :Y Z be any function. Then g o f is R-supercontinuous if and only if g is continuous. Further, if in addition f maps r-open (r-closed) sets to r-open (r-closed) sets, then g is R-supercontinuous. Proof. Sufficiency is immediate in view of Theorem 4.8. To prove ne- cessity, suppose that g o f is -R-supercontinuous. To show that g is con- R-supercontinuous functions 711 tinuous, let W be an open (closed) subset of Z. Then (g o /)-1(W) = /-1(<7_1(W)) is r-open (r-closed) in X. Since / is an i?-open (.R-closed) surjection /(/-1(i/-1(W))) = f'W is an open (closed) set in Y. The last assertion is immediate in view of Theorem 4.1. •
Theorem 4.14. Let f : X —> Y be a function. Then the following state- ments are true. (a) If f is R-supercontinuous and if A is a subspace of X, then the restriction function f\A : A —> X is R-supercontinuous. (b) Let X = (J Ua, where each Ua is an r-open subset of X. If for each aeA a € A, fa = f\Ua is R-supercontinuous, then f is R-supercontinuous. n (c) Let X = (J Fi, where each Fi is an r-closed subset of X. If for each i=1 i = 1,..., n, fi = f\Ui is R-supercontinuous, then f is R-supercontinuous. Proof, (a) Let V be any open set in Y. Since / is -R-supercontinuous. l f~ (V) is an r-open set. Now, since (/|j4)-1 = /-1(V) fl A, it is r-open in A and so f\A is .R-supercontinuous. (b) Let V be an open subset of Y. Then /-100 = Uj/'^F) : a e A}. 1 Now, i?-supercontinuity of fa implies that /QT (V) is r-open in Ua for each a and hence in X. Since arbitrary union of r-open sets is r-open, /_1(Vr) is r-open. (c) Let F be any closed subset of Y. Then f~l(F) = (J /¿_1(F). By ¿=i ii-supercontinuity of fi (i = 1,2,... ,n), f~l(F) is r-closed in Fi and hence in X. Now /_1(F) being a finite union of r-closed sets is r-closed set. So / is .R-supercontinuous. • Our next result shows that /2-supercontinuity is preserved under the shrinking of its range. Theorem 4.15. If f : X —*Y is R-supercontinuous and f(X) is endowed with the subspace topology, then f : X —• f(X) is R-supercontinuous. Proof. Since / is -R-supercontinuous, for every open subset V of Y, f~1(VT\ f(X)) = f~HV) n rH/PO) = rHV) nx = r\V) is r-open. So / : X —* f(X) is .R-supercontinuous. • In contrast to Theorem 4.15, it is easily verified that R-supercontinuity is invariant under the expansion of its range.
Theorem 4.16. Let f : X IlaeA^a be defined by f(x) = (fQ{x))aeA, where fa:X—> Xa is a function for each aeA. Let riaeA be endowed with the product topology. Then f is R-supercontinuous if and only if each fa is R-supercontinuous. 712 J. K. Kohli, D. Singh, J. Aggarwal
Proof. Suppose that the mapping / : X —> f|aeA Xa is i?-supercontinuous. Then fa — na o /, where 7rQ : Xa —• Xa denotes the projection onto the «-coordinate space Xa. Since projection maps are continuous, in view of Theorem 4.8, fa is a .R-supercontinuous for each a.
Conversely, assume that each fa : X —> Xa is i?-supercontinuous. To prove il-supercontinuity of / it suffices to show that inverse image under / of each subbasic open set in the product space HQCA Xa is r-open. Let Upx
YL#Xa be a subbasic open set in Xa. Then Up x X\a^pXa) = aeA f~l{^pl{Up)) = fpl{Up) is r-open in X and so, / is .R-supercontinuous. •
THEOREM 4.17. Let f : IlaeA^a FLeA^* ¿>e a mapping defined by f((xa)) = (fa(xa)), where fa : Xa —> Ya for each a £ A. Then f is R-supercontinuous if and only if each fa is R-supercontinuous.
Proof. To prove necessity, let Vp be any open set in Yp. Then ^~pl{Vp)
= Vpx naz£p Ya is a subbasic open set in HQGA Now since / is -R-super- -1 1 l v x X is r en continuous, Z ^ ^)) = fp ( p) 11a ^p <* "°P in riaeA^«- Thus Z"1 (Vp) is an r-open set in Xp and so fp is .R-supercontinuous.
Conversely, let V = Vp x Ya be a subbasic open set in the product -1 _1 space n^a- Then / (V) = / (Vg x Y\a^pYa). Now since each fa is ii-supercontinuous, /_1(V) is r-open and hence / is .R-supercontinuous. •
THEOREM 4.18. Let f : X —> Y be a function and g : X —> X x Y be the graph function defined by g(x) = (x,f(x)) for each x G X. Then g is R-supercontinuous if and only if f is R-supercontinuous and X is an Ro-space. Proof. Suppose that g is .R-supercontinuous. By Theorem 4.8, / = iry ° g is .R-supercontinuous, where ixy \ XxY Y denotes the projection map. To show that X is an i?o-space, let U be an open set in X and let x € U. Then U x Y is an open set containing g(x). By ii-supercontinuity of g, there is an r-open set W containing x such that g(W) C U x Y. So W C U. Thus U being a union of r-open sets is r-open in X and hence X is an .Ro-space. Conversely, to prove ii-supercontinuity of g, let x £ X and let W be an open set containing g(x). Then there exist open sets U C X and V C Y such that g(x) = (x, f(x)) C U x V C W. Since X is an Ro-space, there exists an r-open set G\ C X such that x G G\ C U. Again by .R-supercontinuity of /, there exists an r-open set G2 in X containing x such that /(G2) C V. Let G = G\ fl G2. Then G is an r-open set containing x such that g(G) C U x V C W, which shows that g is -R-supercontinuous. • R-supercontinuous functions 713
REMARK 4.19. The hypothesis of 'HQ-space' in Theorem 4.18 cannot be omitted. For let X be the real line with the right ray topology [35] and Y be the real line with indiscrete topology. Let / : X —» Y be any function. Clearly / is i?-supercontinuous but the graph function g : X —> X x Y is not .R-supercontinuous. It is well known that the equalizer of two continuous functions into a Hausdorff space is closed. In case of -R-supercontinuous functions we have the following strong version of the same. THEOREM 4.20. Let f,g:X—>Y be R-supercontinuous functions from a topological space X into a Hausdorff space Y. Then the equalizer E = {x G X : f(x) — of the function f and g is an r-closed subset of X. Proof. To prove that E is r-closed, we shall show that its complement X \ E is r-open. To this end, let x G X \ E. Then f(x) ^ g(x). In view of Hausdorffness of Y, there exist disjoint open sets V and W such that f(x) G V and g(x) G W. Since / and g are .R-supercontinuous, /-1(V) and g~l{W) are r-open sets containing x. Then U = /-1(V) fl g~1(W) is an r-open set containing x such that U fl E =
THEOREM 4.21. [19] If f : X Y is R-supercontinuous from an R- compact space X onto a space Y, then Y is compact.
5. Topological properties and i?-supercontinuity THEOREM 5.1. Let f : X Y be an R-supercontinuous open bijection. Then X and Y are homeomorphic RQ- spaces. Proof. Let U be an open set in X and let x € X. Since / is an open map, f(U) is an open set containing f(x). Now since / is .R-supercontinuous, there exists an r-open set G containing x such that f(G) C f(U). Now, x G f~l(f(G)) C f~l{f{U)). Since / is a bijection, f~\f{G)) = G and /-1 (/([/)) = U. So x G G C U. Thus U being a union of r-open sets is r-open and hence X is an iio-space. Since / is a homeormorphism and since the property of being an iio-space is a topological property, Y is an .Ro-space. • THEOREM 5.2. Let f : X —> Y be an R-supercontinuous injection into a To-space Y. Then X is a T\ -space. Further, if X is an Ri -space, then X is Hausdorff.
Proof. Let xi,x2 G X,x\ ^ x2. Then f(xi) ^ f(x2). Since Y is a 7o-space, there exists an open set V containing either f(x 1) or f(x2) but not both. 714 J. K. Kohli, D. Singh, J. Aggarwal
To be precise, suppose that f(xi) 6 V. Then is an r-open set containing xi but not £2• Now xi G {a;i} C /_1(F) and so X — {xi} is an open set containing X2 but not x\. Thus X is a Ti-space. The last assertion is immediate in view of the fact that a To, R\-space is Hausdorff [7]. •
DEFINITION 5.3. [13] A space X is said to be (i) r-regular if for every r-closed set A and a point x outside A, there exist open sets U and V in X such that x € U, A C V and U HV =
THEOREM 5.4. Let X be an r-regular space and let f : X —> Y be an R-supercontinuous open bijection. Then both X and Y are regular spaces. Proof. Let A be a closed subset of Y and let y 0 A. Since / is a bi- jection, /_1(y) is a singleton and /_1(?/) ^ Again, since / is R- supercontinuous f~1(A) is an r-closed set in X. In view of r-regularity of X, there exist disjoint open sets U and V such that f~1(y) G U and f~1(A) C V. Now, since / is an open bijection, f(U) and /(V) are disjoint open sets containing y and A, respectively and so Y is regular. Since / is a homeomorphism and since regularity is a topological property, X is also regular. •
THEOREM 5.5. If f : X —> Y is an R-supercontinuous function and if X is an r-regular space, then f is strongly 6-continuous. Proof. Let x E X and V be an open set containing f(x). Since / is i?-supercontinuous, there exists an r-open set U containing x such that f(U) C V. Again, since X is an r-regular space, there exists an open set W such that x G W C W C U. Now f(W) C f(U) C V and so / is strongly 0-continuous. •
DEFINITION 5.6. A function / : X —> Y is said to be an R-homeomor- phism, if / is a bijection such that / and f~l are ii-supercontinuous.
THEOREM 5.7. Let f : X —> Y be an R-homeomorphism from an r- completely regular space X onto Y. Then both the spaces X and Y are completely regular. Proof. Let A be a closed set in Y and let y £ A. Then x = f~1(y) is a singleton and x does not belongs to the r-closed set f~1(A). In view of the r-complete regularity of X, there exists a continuous function g : X —> [0,1] such that g(x) = 0 and g(f~1(A)) = 1. Let h = g o /-1. Since / is an ii-homeomorphism, h is well defined and since f~l is i?-supercontinuous, R-supercontinuous functions 715 h is continuous. Moreover, h{y) = 0 and h(A) = 1. Thus Y is a com- pletely regular space. Now, since / is a homeomorphism and since complete regularity is a topological invariant, the space X is also completely regular. •
6. Properties of graph of an i?-supercontinuous function Let / : X —> Y be an i?-supercontinuous function. Since every R- supercontinuous function is continuous, the family {1 x>/}, where lx de- notes the identity map on X, separates points and separates points from closed sets. So the mapping g : X —> X x Y defined by g(x) = (x, f{x)) is an embedding of X into X xY. Thus X is homeomorphic to its graph G(f) = g{X). Hence every topological property enjoyed by X is transferred to its graph G(f) and conversely. The next two definitions shed some light on the fact that how the graph of an i?-supercontinuous function / : X —> Y is situated in the product space X x Y.
DEFINITION 6.1. The graph G(f) of / : X —> Y is called (i) r-closed with respect to X if for each (a;, y) £ G(f) there exist open sets U and V containing x and y respectively, such that U is r-open and (U xV) n G(f) =
THEOREM 6.2. If f : X —• Y is R-supercontinuous function and Y is Hausdorff, then G(f) is r-closed with respect to X xY. Proof. Let x G X and let y /(x). Then there exist disjoint open sets V and W containing y and f(x) respectively. Since / is ii-supercontinuous, there is an r-open set U containing x such that f(U) dWdY — V. Then it is easily verified that U x V is an r-open set containing (x, y) such that (U x V) fl G(f) = (f>. Consequently, G(f) is r-closed in X x Y. •
COROLLARY 6.3. If f : X —> Y is R-supercontinuous and Y is Hausdorff, then G(f), the graph of f is r-closed with respect to X.
7. r-Quotient topology and r-quotient spaces Let / : X —> Y be a surjection defined on a topological space X to a set Y. The largest topology on Y which makes / continuous is called the quotient topology on Y. Several variants of quotient topology occur in the mathematical literature which in general are coarser than the quotient topology but coincide with it if X is suitably augmented. For example, see ([14], [15], [16], [27], [32], [33]). The interrelations and interplay among 716 J. K. Kohli, D. Singh, J. Aggarwal certain of these variants of quotient topology are well discussed in [20]. In this section, we introduce a new variant of quotient topology called 'r-quotient topology'' and elaborate on its interrelations with other variants of quotient topology that already exist in the literature. DEFINITION 7.1. Let p : X —> Y be a function from a topological space X onto a set Y. The collection of all subsets A c Y such that p~l(A) is a (1) r-open set in X is a topology on Y and is called the r-quotient topology. The map p is called the r-quotient map and the space Y with r-quotient topology is called r-quotient space. (2) d-open set in X is a topology on Y and is called the cl-quotient topology [33]. The map p is called the cl-quotient map. (3) z-open set in X is a topology on Y and is called the z-quotient topology [14]. The map p is called the z-quotient map. (4) «¿¿-open set in X is a topology on Y and is called the Dg-quotient topology [16]. The map p is called the Ds-quotient map. (5) cT-open in X is a topology on Y and is called the D*-quotient topology [32], The map p is called the D*-quotient map. (6) d-open set in X is a topology on Y and is called the D- quotient topology [15]. The map p is called the D- quotient map. (7) 0-open set in X is a topology on Y and is called the 6-quotient topology [20]. The map p is called the 6-quotient map. (8) F-open set in X is a topology on Y and is called the F-quotient topology [18]. The map p is called the F-quotient map. (9) ¿-open set in X is a topology on Y and is called the 6-quotient topology [27]. The map p is called the 6-quotient map. The following diagram well reflects upon the place of r-quotient topology in the hierarchy of other variants of quotient topology that already exist in the literature.
cl-quotient topology c z-quotient topology cz F-quotient topology D'-quotient topology D5-quotient topology quotient topology n n n D-quotient topology 0-quotient topology n n r-quotient topology n quotient topology However, none of the above inclusions is reversible in general as is shown by examples in ([14], [15], [16], [20]) and the following examples. R-supercontinuous functions 717 EXAMPLE 7.2. Let X — Y be the set of all real numbers and let X be endowed with the right ray topology r. Let / denote the identity map from X onto Y. Then the quotient topology on Y is identical with r while r-quotient topology on Y is the indiscrete topology. EXAMPLE 7.3. Let (X,r) denote the mountain chain space due to Helder- mann [11], Let Y — X and / be the identity map from X onto Y. Since (X, r) is a regular space which is not D-regular, the r-quotient topology and the ^-quotient topology on Y are identical with r. However, the D-quotient topology on Y is strictly coarser than r. EXAMPLE 7.4. Let X = Y be the set of natural numbers. Let X be endowed with the cofinite topology rc and let / denote the identity map defined on X. Then the r-quotient topology on Y is identical with rc while ^-quotient topology on Y is indiscrete. THEOREM 7.5. Let p : X —• Y be a surjection from a topological space (X,TI) onto a topological space (Y,72), where is the r-quotient topology. Then p is R-supercontinuous. Moreover, T2 is the largest topology on Y which makes p : X —> Y R-supercontinuous. Proof. The fact that p is .R-supercontinuous follows from the definition of r-quotient topology. To prove the last assertion, let T* be a topology on Y such that p : (X, T\) —> (Y,T*) is .R-supercontinuous. Let V be a r*-open set in Y. Then since p is .R-supercontinuous, p~l (V) is r-open in X. Now by definition of r-quotient topology, V is T2-open and so r* C • The following result shows that a function out of r-quotient space is continuous if and only if its composition with r-quotient map is il-super- continuous. THEOREM 7.6. Let p : X —• Y be an r-quotient map. Then a function g :Y —> Z is continuous if and only if g op is R-supercontinuous. Proof. Suppose that g o p is .R-supercontinuous and let W be an open set in Z. Then (g op)_1(W) = p~1(g~1(W)) is r-open in X. Since p is an r-quotient map, g~l{W) is open in Y. Thus g is continuous. Converse is immediate in view of definition of r-quotient topology. • 8. Strong variants of continuity and change of topology Let / : X —> Y be a function from a topological space X into a topological space Y satisfying one of the strong forms of continuity of Definition 2.1 (a)-(j). If the domain of / is retopologized in an appropriate way, then / is simply a continuous function. Mathematical literature is replete with examples of this type. For example, see ([5], [11], [14], [15], [16], [24], [33]). In this section, we retopologize the domain of an .R-supercontinuous function 718 J. K. Kohli, D. Singh, J. Aggarwal in such a way that it is simply a continuous function. Moreover, we elaborate on the interrelations and interplay among various coarser topologies obtained in this way for a given topology. As a byproduct we conclude with alternative proofs of certain results of preceding sections. 8.1. Let (X, r) be a topological space. Let (i) Br denote the set of all r-open subsets of the space (X, r). Since arbitrary union and finite intersections of r-open sets is r-open, the collection Br is indeed a topology for X. We shall denote this topology by tr. Clearly rR C r. (ii) Bz denote the collection of all cozero subsets of the space (X, r). Since the intersection of the two cozero sets is a cozero set, the collection Bz is a base for a topology TZ on X [14]. The topology TZ has been referred to by several authors in the literature. See for example, Aull [3], D'Aristotle [6] and Stephenson [34], (iii) Bds denote the collection of all regular iv-sets, the collection Bd6 is a base for a topology r^ on X [16]. (iv) Bq denote the collection of 0-open sets of the space (X,t). Since ar- bitrary union and finite intersections of #-open sets are 0-open, Bq is indeed a topology on X. We shall denote this topology by tq. The topology tq has been extensively referred to in the literature (see [24], [36]). (v) Bs denote the collection of all regular open subsets of the space (X, r). Since the intersection of two regular open sets is regular open, the collection Bs constitutes a base for a topology t$ on X and is called semiregular topology associated with r. The space (X, T$) is often called the semiregularization of the space and has been extensively used in the topological literature (see [5, Exercise 20, p. 138], [8]). (vi) Bd denote the collection of all open F^-subsets of the space (X, r). Since the intersection of two open Fa-sets is an open FCT-set, the collection Bd is a base for a topology r^ on X ([11], [15]). (vii) Bd* denote the collection of all strongly open F^-sets of the space (X, r). Since the intersection of two strongly open FCT-sets is a strongly open FCT-set, the collection Bd* constitutes a base for a topology Td* on X ([11], [32])- (viii) Bf denote the collection of all F-open subsets of the space (X, r). Since arbitrary union and finite intersection of F-open sets is F-open, the collection Bf is indeed a topology on X denoted by tf [18]. (ix) Bd denote the collection of all cl-open subsets of the space (X, r). Since the intersection of two cl-open sets is cl-open. The collection Bd constitutes a base for a topology rc; on X ([9], [33]). R-supercontinuous functions 719 The following diagram well illustrates the interrelations that exist among various coarser topologies on the space (X, r) elucidated in the above para- graphs. Td n D Tz C Tf C TR n n TRD TD D TDS n r D TS D TE C TR C r However, in general none of the above inclusions is reversible as is well reflected by the existing examples in the mathematical literature and the following examples. EXAMPLES 8.2. Let (X, r) denote the real line endowed with the usual Euclidean topol- ogy. Then r2 = r and Td is the indiscrete topology. 8.3. Let (X, r) be the space of irregular lattice topology [35, Example 79, p. 97]. This space is a functionally regular space which is not a semiregular space. The topology rp is identical with r but tz is strictly coarser than T. Again the topology TS is strictly weaker than r. Moreover, TR = T and TQ is strictly weaker than r. 8.4. Let (X,T) be the regular space due to Hewitt [12] on which every continuous real valued function is constant. Then TR is identical with r but Tp is the indiscrete topology. Moreover, tq = r and Tds is strictly coarser than r. 8.5. Let (X,T) denote the space of [17, Example 3.3]. The space (X,T) is a functionally Hausdorff /^-completely regular space which is not a completely regular space. Then Tds is same as r but r2 is strictly coarser than r. 8.6. Let (X, r) denote the space of [17, Example 3.6]. This space is a regular space as well as a D-regular space but not a Z)j-completely regular space. In this case tq = Td = r but Tds is strictly coarser than r. 8.7. Let (X, T) denote the skyline space due to Heldermann [11]. Then X is a regular space as well as a D-regular space which is not a £)-completely regular space. Then r^ = TQ = r and r^» is strictly coarser than r. 8.8. Let (X, r) denote the set of integers endowed with the cofinite topology. Then (X, r) is a D-completely regular space which is not regular. So TD* = T but TZ = TQ is the indiscrete topology. 720 J. K. Kohli, D. Singh, J. Aggarwal THEOREM 8.9. A function f : (X, r) —> (Y, v) is R-supercontinuous if and only if f : (X, TR) —> (Y, V) is continuous. Many of the results in the preceding sections now follow from Theorem 8.9 and corresponding standard properties of continuous functions. THEOREM 8.10. For a topological space {X,T) the following statements are equivalent. (a) (X,T) is an RO-space. (b) Every continuous function from (X, r) into a space (Y, u) is R-super- continuous. Proof. (a)=>(b) is obvious. (b)=>(a) Take (Y, v) = (X, r). Then identity function LX on X is contin- uous and hence /¿-supercontinuous. So by above Theorem lx : (X,TR) —> (X,T) is continuous. Since U € r implies 1~1({7) = U E TR, therefore r C TR. Thus it follows that T — TR and hence (X, r) is an i?o-space. • DEFINITION 8.11. [21] A function / : X —> Y from a topological space X into a topological space Y is said to be R-continuous at x 6 X if for each r-open set V containing f(x) there exists an open set U containing x such that f(U) C V. The function / is said to be R-continuous if it is /¿-continuous at each point of X. THEOREM 8.12. Let f : (X,T) (Y,V) be a function. Then (a) / is R-continuous if and only if f : (X, r) —» (Y, UR) is continuous. (b) / is R-open if and only if f : (X, TR) —> (Y, v) is open. In view of Theorems 8.9 and 8.10, Theorem 4.13 can be restated as follows: If / : (X, TR) (Y, v) is a continuous open surjection and g : (Y, u) —» (Z, a) is a function, then g is continuous if and only if g o f is continuous. Further, if / maps open (closed) sets to r-open (r-closed) sets, then g is i2-supercontinuous. Moreover, r-quotient topology on Y determined by the surjection / : (X, T) —> Y in Section 7 is identical with the standard quotient topology on Y determined by / : (X, TR) —> Y. 9. Appendix (The category UPS) Details of the concepts and definitions in this section may be found in [10]. However, for the convenience of the reader we include here the basics of the category UPS which is a full subcategory of TOP (= the category of topological spaces and continuous maps). R-supercontinuous functions 721 Let L be a nonempty set equipped with a transitive relation <. We say that a is a lower bound of a set X C L, and b is an upper bound if a < x for each x G X, and x < b for each x G X, respectively. If the set of all upper bounds of X has the smallest element, we call this the supremum of X (sup X). Similarly, if the set of all lower bounds of X has the largest element, we call this the infimum of X (inf X). The set L is said to be directed if every finite subset of L has an upper bound in L. Let ]X = {y G L : x < y for some x G X}. A set X C L is said to be an upper set ifX = A partially ordered set or poset is a nonempty set L equipped with a reflexive, transitive and antisymmetric relation. A complete lattice is a poset in which every subset has a supremum and infimum. Let L be a complete lattice and let U C L. The set U is said to be Scott open if (i) U is an upper set and (ii) sup D G U implies U fl D ^ Acknowledgement Authors are thankful to the referee for helpful suggestions which led to the improvement of an earlier version of the paper and inclusion of Appendix (Section 9). References [1] P. Alexandroff, Discrete raiime, Mat. Sb. 2 (1937), 501-518. [2] C. E. Aull, Notes on separation by continuous functions, Indag. Math. (N.S.) 31 (1969), 458-461. [3] C. E. Aull, On C-and C*-embeddings, Indag. Math. (N.S.) 37 (1975), 26-33. [4] C. E. Aull, Functionally regular spaces, Indag. Math. (N.S.) (1976), 281-288. [5] N. Bourbaki, General Topology Part I, Hermann, Addison-Wesley (1966). [6] A. J. D'Aristotle, Quasicompact and functionally Hausdorff spaces, J. Austral. Math. Soc. 15 (1973), 319-324. [7] A. S. Davis, Indexed system of neighbourhoods for general topological spaces, Amer. Math. Monthly 68 (1961), 886-893. 722 J. K. Kohli, D. Singh, J. Aggarwal [8] R. F. Dickman Jr., J. R. Porter, 6-closed subsets of Hausdorff spaces, Pacific. J. Math. 59 (1975), 407-415. [9] E. Ekici, Generalizations of perfectly continuous, regular set connected and clopen functions, Acta Math. Hungar. 107(3) (2005), 193-205. [10] G. Gierz, K. H. Hoffman, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott, A Compendium of Continuous Lattices, Springer Verlag, Berlin (1980). [11] N. C. Heldermann, Developability and some new regularity axioms, Canad. J. Math. 33(3) (1981), 641-663. [12] E. Hewitt, On two problems of Urysohn, Ann. of Math 47(3) (1946), 503-509. [13] J. K. Kohli, Variants of (complete) regularity and factorizations of (complete) regu- larity (preprint). [14] J. K. Kohli, R. Kumar, z-supercontinuous functions, Indian J. Pure Appl. Math. 33(7) (2002), 1097-1108. [15] J. K. Kohli, D. Singh, D-supercontinuous functions, Indian J. Pure Appl. Math. 32(2) (2001), 227-235. [16] J. K. Kohli, D. Singh, D¡-supercontinuous functions, Indian J. Pure Appl. Math. 34(7) (2003), 1089-1100. [17] J. K. Kohli, D. Singh, Between regularity and complete regularity and a factorization of complete regularity, Stud. Cere. Mat. Vol. (17) (2007), 125-134. [18] J. K. Kohli, D. Singh, J. Aggarwal, F-supercontinuous functions, Applied General Topology 10(1) (2009), 69-83. [19] J. K. Kohli, D. Singh, J. Aggarwal, On two new factorizations of compactness, preprint. [20] J. K. Kohli, D. Singh, R. Kumar, Some properties of strongly 6-continuous functions, Bull. Calcutta Math. Soc. 100 (2008), 185-196. [21] J. K. Kohli, D. Singh, R. Kumar, J. Aggarwal, Between continuity and set connect- edness, Applied General Topology (to appear). [22] N. 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Shanin, On separation in topological spaces, Dokl. Akad. Nauk. SSSR 38 (1943), 110-113. [32] D. Singh, D*-supercontinuous functions, Bull. Calcutta Math. Soc. 94(2) (2002), 67-76. R-supercontinuous functions 723 [33] D. Singh, cl-supercontinuous functions, Applied General Topology 8(2) (2007), 293-300. [34] R. M. Stephenson, Spaces for which Stone-Weierstrass theorem holds, Trans. Amer. Math. Soc. 133 (1968), 537-546. [35] L. A. Steen, J. A. Seeback, Jr., Counter Examples in Topology, Springer Verlag, New York, 1978. [36] N. H. Velicko, H-closed topological spaces, Amer. Math Soc. Transi. 78(2) (1968), 103-118. J. K. Kohli DEPARTMENT OF MATHEMATICS HINDU COLLEGE UNIVERSITY OF DELHI DELHI 110007, INDIA E-mail: [email protected] D. Singh DEPARTMENT OF MATHEMATICS SRI AUROBINDO COLLEGE UNIVERSITY OF DELHI NEW DELHI 110017, INDIA E-mail: [email protected] J. Aggarwal DEPARTMENT OF MATHEMATICS UNIVERSITY OF DELHI DELHI 110007, INDIA E-mail: [email protected] Received February 13, 2009; revised version July 7, 2009.