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A new generalization of contra-continuity via Levine’s g-closed sets
Chaos, Solitons and Fractals 32 (2007) 1597–1603 www.elsevier.com/locate/chaos
A new generalization of contra-continuity via Levine’s g-closed sets Miguel Caldas
a,*
, Saeid Jafari b, Takashi Noiri c, Marilda Simo˜es
d
a
Departamento de Matematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos Braga, s/n 24020-140, Niteroi, RJ, Brazil b College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark c Yatsushiro College of Technology, 2627 Hirayama shinmachi, Yatsushiro-shi, Kumamoto-ken 866-8501, Japan d Dipartimento Di Matematica ‘‘Guido Castelnuovo’’, Universita´ Di Roma ‘‘La Sapienza’’, 00185 Roma, Italy Accepted 13 December 2005
Abstract In [Dontchev J. Contra-continuous functions and strongly S-closed spaces. Int J Math Math Sci 1996;19:303–10], Dontchev introduced and investigated a new notion of continuity called contra-continuity. Recently, Jafari and Noiri [Jafari S, Noiri T. Contra-a-continuous functions between topological spaces. Iran Int J Sci 2001;2:153–67, Jafari S, Noiri T. Contra-super-continuous functions. Ann Univ Sci Budapest 1999;42:27–34, Jafari S, Noiri T. On contraprecontinuous functions. Bull Malaysian Math Sci Soc 2002;25(2):115–28] introduced new generalizations of contracontinuity called contra-a-continuity, contra-super-continuity and contra-precontinuity. In this paper, we introduce and investigate a generalization of contra-continuity by utilizing Levine’s generalized closed sets. 2006 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries General Topology has shown its fruitfulness in both the pure and applied directions. In data mining [28], computational topology for geometric design and molecular design [24], computer-aided geometric design and engineering design (briefly CAGD), digital topology, information systems, non-commutative geometry and its application to particle physics [19], one can observe the influence made in these realms of applied research by general topological spaces, properties and structures. Rosen and Peters [29] have used topology as a body of mathematics that could unify diverse areas of CAGD and engineering design research. They have presented several examples of the application of topology to CAGD and design. The concept of closedness is fundamental with respect to the investigation of general topological spaces. Levine [20] initiated the study of the so-called g-closed set and by doing this, he generalized the concept of closedness. The concept of g-closed sets were also considered by Dunham [11] in 1982 and by Dunham and Levine [9] in 1980. Balachandran et al. [2] in 1991, defined a new class of mappings called generalized continuous (briefly g-continuous) mappings which *
Corresponding author. E-mail addresses: [email protected]ff.br (M. Caldas), [email protected] (S. Jafari), [email protected] (T. Noiri), [email protected] (M. Simo˜es). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.12.032
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contains the class of continuous mappings. Since the advent of these notions, several research papers with interesting results in different respects came to existence (see, [2–4,9–11,22]). Recently, Jafari and Noiri introduced and investigated the notions of contra-precontinuity [18], contra-a-continuity [16] and contra-super-continuity [17] as a continuation of research done by Dontchev [7] and Dontchev and Noiri [8] on the interesting notions of contra-continuity and contrasemi-continuity, respectively. Caldas and Jafari [6] introduced and investigated the notion of contra-b-continuous functions in topological spaces as a new generalization of both contra-semi-continuity and contra-precontinuity. The present note has as its purpose to investigate some fundamental properties of contra g-continuous functions by using g-open sets. We believe that the notions introduced in this paper have applications in a broader sense. For example one can talk about g-manifolds in which everything open is changed by g-open sets such as Hausdorffness by g-T2 (defined as Hausdorff but by changing open with g-open), also continuity by contra g-continuity or g-continuity. We know that manifolds are fundamental in many branches of science in general and mathematical physics in special. On g-manifolds, one can impose GO-compactness [2] which its definition is similar to the definition of compactness by changing open with gopen. Then one can talk about Riemannian g-manifolds and semi-Riemannian g-manifolds. Indeed one can also talk about g-paracompact manifolds by using g-open sets instead of open sets in paracompact manifolds. The same is true with the definition of g-orbifold and lots of other geometrical objects. Moreover the notion of kernel of a set has applications in computer science [30]. This notion is used in Theorem 2.2 and Corollary 2.3 of this paper. It should be mentioned that the present work may become relevant to the work of Witten [32] and El-Naschie [13,14]. Also the fuzzy topological version of the notions and results introduced in this paper are very important since Professor El-Naschie has recently shown in [15] that the notion of fuzzy topology my be relevant to quantum particle physics in connection with string theory and e1 theory. Throughout this paper, (X, s) and (Y, r) (or X and Y) always denote topological spaces. A subset A of X is said to be regular open (resp. regular closed) if A = Int(Cl(A)) (resp. A = Cl(Int(A))) where Cl(A) and Int(A) denote the closure and interior of A, respectively. A subset A of a space X is called preopen [23] (resp. semi-open [21], a-open [27], b-open [1]) if A Int(Cl(A)) (resp. A Cl(Int(A)), A Int(Cl(Int(A))), A Cl(Int(Cl(A)))). The complement of a preopen (resp. semi-open, a-open, b-open) set is said to be preclosed (resp. semi-closed, a-closed, b-closed). The collection of all closed (resp. semi-open, clopen) subsets of X will be denoted by C(X) (resp. SO(X), CO(X)). We set C(X, x) = {V 2 C(X)/x 2 V} for x 2 X. We define similarly CO(X, x). A subset B of a topological space X is called g-closed in X [20] if Cl(B) G whenever B G and G is open in X. The union of two g-closed sets is a g-closed set. A subset A is called g-open in X if its complement, X A = Ac is g-closed. The intersection of all g-closed sets containing a set A is called the g-closure of A [11] and is denoted by Cl*(A). This is, for any A X, Cl*(A) = \{F X : A F and F is g-closed}. If A X, then A Cl*(A) Cl(A). The collection of all g-closed (resp. g-open) subsets of X will be denoted by GC(X) (resp. GO(X)). We set GC(X, x) = {V 2 GC(X)/x 2 V} for x 2 X. We define similarly GO(X, x). If A is g-open in X and B is g-open in Y, then A · B is g-open in X · Y [20]. Let p be a point of X and N be a subset of X. N is called a g-neighborhood of p in X [3] if there exists a g-open set O of X such that p 2 O N. Lemma 1.1 [3]. Let A be a subset of X. Then, p 2 Cl*(A) if and only if for any g-neighborhood Np of p in X, A \ Np 5 /. A mapping f : X ! Y from a topological space X into a topological space Y is called g-continuous [2] if the inverse image of every closed set in Y is g-closed in X. Observe that a mapping f : X ! Y is g-continuous if and only if the inverse image of every open set in Y is g-open in X.
2. Contra g-continuous functions Definition 1. A function f : X ! Y is said to be contra g-continuous if f1(V) is g-closed in X for each open set V of Y. Definition 2. Let A be a subset of a space (X, s). The set \{U 2 sjA U} is called the kernel of A [25] and is denoted by ker(A). Lemma 2.1 [17]. The following properties hold for subsets A, B of a space X: (1) x 2 ker(A) if and only if A \ F 5 ; for any F 2 C(X, x). (2) A ker(A) and A = ker(A) if A is open in X. (3) If A B, then ker(A) ker(B).
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Theorem 2.2. Let f : X ! Y be a function from a topological space X into a topological space Y. (i) The following statements are equivalent. (a) f is contra g-continuous. (b) The inverse image of each closed set in Y is g-open in X. (ii) The following statements are equivalent. (a) For each point x in X and each closed set V in Y with f(x) 2 V, there is a g-open set U in X containing x such that f(U) V. (b) For every subset A of X, f(Cl*(A)) ker(f(A)) holds. (c) For each subset B of Y, Cl*(f1(B)) f1(ker(B)). Proof (i) (a) M (b): see Definition 1. (ii) (a) ! (b): Let A be any subset of X. Suppose that y 62 ker(f(A)). Then, by Lemma 2.1 there exists V 2 C(Y, y) such that f(A) \ V = ;. For any x 2 f1(V), by (a) there exists Ux 2 GO(X, x) such that f(Ux) V. Hence f(A \ Ux) f(A) \ f(Ux) f(A) \ V = ; and A \ Ux = ;. This shows that x 62 Cl*(A) for any x 2 f1(V). Therefore, f1(V) \ Cl*(A) = ; and hence V \ f(Cl*(A)) = ;. Thus, y 62 f(Cl*(A)). Consequently, we obtain f(Cl*(A)) ker(f(A)). (b) ! (c): Let B be any subset of Y. By (b) and Lemma 2.1, we have f(Cl*(f1(B))) ker(ff1(B)) ker(B) and Cl*(f1(B)) f1(ker(B)). (c) ! (a): Suppose that (c) holds. Let x 2 X and V 2 C(Y, f(x)). By (c) and Lemma 2.1, Cl*(f1(YnV)) f1(ker(YnV)) = f1(YnV) and hence Cl*(f1(YnV)) = f1(YnV). Since x 2 Xnf1(V), there exists U 2 GO(X, x) such that U \ f1(YnV) = ;. Therefore, we obtain f(U) V. h Corollary 2.3. Assume that Cl*(A) is g-closed for each subset A of X. Then the following statements are equivalent for a function f : X ! Y: (1) (2) (3) (4) (5)
f is contra g-continuous, for every closed subset F of Y, f1(F) 2 GO(X), for each x 2 X and each F 2 C(Y, f(x)), there exists U 2 GO(X, x) such that f(U) F, f(Cl*(A)) ker(f(A)) for every subset A of X, Cl*(f1(B)) f1(ker(B)) for every subset B of Y.
Proof. The implications (1) ! (2) and (2) ! (3) are obvious. (3) ! (4) ! (5): Theorem 2.2(ii). (5) ! (1): Let V be any open set of Y. Then, by Lemma 2.1 we have Cl*(f1(V)) f1(ker(V)) = f1(V) and Cl*(f1(V)) = f1(V). This shows that f1(V) is g-closed in X [3, Remark 4]. h Remark 2.4. The following two examples show that g-continuity and contra g-continuity are independent concepts. Example 2.5. The identity function on the real line (with the usual topology) is continuous and hence g-continuous but not contra g-continuous, since the preimage of each singleton fails to be g-open. Example 2.6. Let X = {a, b} be the Sierpinski space with the topology s = {;, {a}, X}. Let f : X ! X be defined by f(a) = b and f(b) = a. Since the inverse image of every open set is g-closed, then f is contra g-continuous, but f1({a}) is not g-open in (X, s). Therefore f is not g-continuous. Definition 3. A function f : X ! Y is said to be contra-continuous [7] (resp. contra-a-continuous [16], contra-precontinuous [18], contra-semi-continuous [8], contra-b-continuous [6] if for each open set V of Y, f1(V) is closed (resp. a-closed, preclosed, semi-closed, b-closed) in X.
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For the functions defined above, we have the following implications: contra-continuity # contra g-continuity
! contra-a-continuity # contra-semi-continuity
! contra-precontinuity # ! contra-b-continuity
Remark 2.7. It should be mentioned that none of these implications is reversible as shown by the examples stated below. Example 2.8 [16]. Let X = {a, b, c}, s = {;, {a}, X} and r = {;, {b}, {c}, {b, c}, X}. Then the identity function f : (X, s) ! (X, r) is contra-a-continuous but not contra-continuous. Example 2.9. Let X = {a, b, c}, s = {;, {a}, X} and r = {;, {b}, {c}, {b, c}, X}. Then the identity function f : (X, s) ! (X, r) is contra g-continuous but not contra-continuous. Example 2.10 [8]. A contra-semi-continuous function need not be contra-precontinuous. Let f : R ! R be the function f(x) = [x], where [x] is the Gaussian symbol. If V is a closed subset of the real line, its preimage U = f1(V) is the union of the intervals of the form [n, n + 1], n 2 Z; hence U is semi-open being union of semi-open sets. But f is not contraprecontinuous, since f1(0.5, 1.5) = [1, 2) is not preclosed in R. Example 2.11 [8]. A contra-precontinuous function need not be contra-semi-continuous. Let X = {a, b}, s = {;, X} and r = {;, {a}, X}. The identity function f : (X, s) ! (Y, r) is contra-precontinuous as only the trivial subsets of X are open in (X, s). However, f1({a}) = {a} is not semi-closed in (X, s); hence f is not contra-semi-continuous. Theorem 2.12. Assume that Cl*(A) is g-closed for each subset A of X. If a function f : X ! Y is contra g-continuous and Y is regular, then f is g-continuous. Proof. Let x be an arbitrary point of X and V an open set of Y containing f(x). Since Y is regular, there exists an open set W in Y containing f(x) such that Cl(W) V. Since f is contra g-continuous, so by Corollary 2.3, there exists U 2 GO(X, x) such that f(U) Cl(W). Then f(U) Cl(W) V. Hence, f is g-continuous. h Definition 4. A space (X, s) is said to be: (1) locally g-indiscrete if every g-open set of X is closed in X, (2) g-space if every g-open set of X is open in X, (3) gS-space if and only if every g-open subset of X is semi-open. The following theorem follows immediately Definition 4: Theorem 2.13. If a function f : X ! Y is contra g-continuous and X is a gS-space (resp. g-space, locally g-indiscrete), then f is contra-semi-continuous (resp. contra-continuous, continuous). Recall that a topological space X is said to be (1) g-T2 [5] if for each pair of distinct points x and y in X there exist U 2 GO(X, x) and V 2 GO(X, y) such that U \ V = ;. (2) Ultra Hausdorff [31] if for each pair of distinct points x and y in X there exist U 2 CO(X, x) and V 2 CO(X, y) such that U \ V = ;. Observe that in the following theorem and its two corollaries, we will assume that Cl*(A) is g-closed for every subset A of a topological space. Theorem 2.14. If X is a topological space and for each pair of distinct points x1 and x2 in X there exists a function f of X into a Urysohn space Y such that f(x1) 5 f(x2) and f is contra g-continuous at x1 and x2, then X is g-T2.
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Proof. Let x1 and x2 be any distinct points in X. Then by hypothesis, there is a Urysohn space Y and a function f : X ! Y, which satisfies the conditions of the theorem. Let yi = f(xi) for i = 1, 2. Then y1 5 y2. Since Y is Urysohn, there exist open neighbourhoods V y 1 and V y 2 of y1 and y2 respectively in Y such that ClðV y 1 Þ \ ClðV y 2 Þ ¼ ;. Since f is contra g-continuous at xi, there exists a g-open neighbourhood U xi of xi in X such that f ðU xi Þ ClðV y i Þ for i = 1, 2. Hence we get U x1 \ U x2 ¼ ; because ClðV y 1 Þ \ ClðV y 2 Þ ¼ ;. Then X is g-T2. h Corollary 2.15. If f is a contra g-continuous injection of a topological space X into a Urysohn space Y, then X is g-T2. Proof. For each pair of distinct points x1 and x2 in X, f is a contra g-continuous function of X into a Urysohn space Y such that f(x1) 5 f(x2) because f is injective. Hence by Theorem 2.14, X is g-T2. h Corollary 2.16. If f is a contra g-continuous injection of a topological space X into a Ultra Hausdorff space Y, then X is g-T2. Proof. Let x1 and x2 be any distinct points in X. Then since f is injective and Y is Ultra Hausdorff, f(x1) 5 f(x2) and there exist V1, V2 2 CO(Y) such that f(x1) 2 V1, f(x2) 2 V2 and V1 \ V2 = ;. Then xi 2 f1(Vi) 2 GO(X) for i = 1, 2 and f1(V1) \ f1(V2) = ;. Thus X is g-T2. h Lemma 2.17 [20]. If Ai is a g-open set in a topological space Xi, for i = 1, 2, . . . , n, then A1 · · An is also g-open in the product space X1 · · Xn. Theorem 2.18. Let f1 : X1 ! Y and f2 : X2 ! Y be two functions, where (1) GO(X1 · X2) is closed under ordinary unions, (2) Y is a Urysohn space, (3) f1 and f2 are contra g-continuous. Then {(x1, x2)/f1(x1) = f2(x2)} is g-closed in the product space X1 · X2. Proof. Let A denote the set {(x1, x2)/f1(x1) = f2(x2)}. In order to show that A is g-closed, we show that (X1 · X2)nA is gopen. Let (x1, x2) 62 A. Then f1(x1) 5 f2(x2) . Since Y is Urysohn, there exist open sets V1 and V2 containing f1(x1) and f2(x2), respectively such that Cl(V1) \ Cl(V2) = ;. Since fi (i = 1, 2) is contra g-continuous, fi1 ðClðV i ÞÞ is a g-open set containing xi in Xi(i = 1, 2). Hence by Lemma 2.17 f11 ðClðV 1 ÞÞ f21 ðClðV 2 ÞÞ is g-open. Further ðx1 ; x2 Þ 2 f11 ðClðV 1 ÞÞ f21 ðClðV 2 ÞÞ ðX 1 X 2 Þ n A. It follows that (X1 · X2)nA is g-open. Thus A is g-closed in the product space X1 · X2. h Corollary 2.19. Let GO(X · X) be closed under unions. If f : X ! Y is contra g-continuous and Y is a Urysohn space, then A = {(x1, x2)/f(x1) = f(x2)} is g-closed in the product space X · X. Definition 5. A topological space X is said to be (1) g-normal if each pair of non-empty disjoint closed sets can be separated by disjoint g-open sets. (2) Ultra normal [31] if each pair of non-empty disjoint closed sets can be separated by disjoint clopen sets. Theorem 2.20. If f : X ! Y is a contra g-continuous, closed injection and Y is ultra normal, then X is g-normal. Proof. Let F1 and F2 be disjoint closed subsets of X. Since f is closed and injective, f(F1) and f(F2) are disjoint closed subsets of Y. Since Y is ultra normal, f(F1) and f(F2) are separated by disjoint clopen sets V1 and V2, respectively. Hence Fi f1(Vi), f1(Vi) 2 GO(X) for i = 1, 2 and f1(V1) \ f1(V2) = ;. Thus X is g-normal. h Theorem 2.21. Let GO(X) be closed under unions. Let f : X ! Y be a function and g : X ! X · Y the graph function, given by g(x) = (x, f(x)) for every x 2 X. Then f is contra g-continuous if and only if g is contra g-continuous. Proof. Let x 2 X and W be a closed subset of X · Y containing g(x). Then W \ ({x} · Y) is closed in {x} · Y containing g(x). Also {x} · Y is homeomorphic to Y. Hence {y 2 Y : (x, y) 2 W} is a closed subset of Y. Since f is contra
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g-continuous, [{f1(y) : (x, y) 2 W} is a g-open subset of X . Further x 2 [{f1(y): (x, y) 2 W} g1(W). Hence g1(W) is g-open. Then g is contra g-continuous. Conversely, let F be a closed subset of Y. Then X · F is a closed subset of X · Y. Since g is contra g-continuous, g1(X · F) is a g-open subset of X. Also g1(X · F) = f1(F). Hence f is contra g-continuous. h Theorem 2.22. If f : X ! Y is a contra g-continuous function and g : Y ! Z is a continuous function, then g f : X ! Z is contra g-continuous. Recall that, a topological space X is said to be T1/2-space [20] if every g-closed set is closed in X. Theorem 2.23. Let X and Z be any topological spaces and Y be a T1/2-space. If f : X ! Y is contra g-continuous and g : Y ! Z is g-continuous, then g f : X ! Z is contra g-continuous. Proof. It follows from definitions.
h
Q g-continuous function. Theorem 2.24. Let {Xk : k 2 X} be any family of topological spaces. If f : X ! X k is a contra Q Then Prk f : X ! Xk is contra g-continuous for each k 2 X, where Prk is the projection of X k onto Xk. Q X k . Since Proof. We shall consider a fixed k 2 X. Suppose Uk is an arbitrary open set in Xk. Then Pr1 k ðU k Þ is open in 1 f is contra g-continuous, we have by definition f 1 ðPr1 k ðU k ÞÞ ¼ ðPrk f Þ ðU k Þ is g-closed in X. Therefore Prk f is contra g-continuous. h Recall that for a function f : X ! Y, the subset {(x, f(x)) : x 2 X} X · Y is called the graph of f and is denoted by G(f). Definition 6. A graph G(f) of a function f : X ! Y is said to be contra g-closed if for each (x, y) 2 (X · Y)nG(f), there exist U 2 GO(X) containing x and V 2 C(Y) containing y such that (U · V) \ G(f) = ;. Lemma 2.25. A graph G(f) of a function f : X ! Y is contra g-closed in X · Y if and only if for each (x, y) 2 (X · Y)nG(f), there exist U 2 GO(X) containing x and V 2 C(Y) containing y such that f(U) \ V = ;. Theorem 2.26. If f : X ! Y is contra g-continuous and Y is Urysohn, then G(f) is contra g-closed in X · Y. Proof. Let (x, y) 2 (X · Y)nG(f), then f(x) 5 y and there exist open sets V, W such that f(x) 2 V, y 2 W and Cl(V) \ Cl(W) = ;. Since f is contra g-continuous, there exists U 2 GO(X, x) such that f(U) Cl(V). Therefore, we obtain f(U) \ Cl(W) = ;. This shows that G(f) is contra g-closed in X · Y. h Definition 7. A topological space X is said to be g-connected [2] if X cannot be expressed as the union of two disjoint non-empty g-open subsets of X. Theorem 2.27. A contra g-continuous image of a g-connected space is connected. Proof. Let f : X ! Y be a contra g-continuous function of a g-connected space X onto a topological space Y. If possible, let Y be disconnected. Let A and B form a disconnection of Y. Then A and B are clopen and Y = A [ B where A \ B = ;. Since f is a contra g-continuous function, X = f1(Y) = f1(A [ B) = f1(A) [ f1(B), where f1(A) and f1(B) are non-empty g-open sets in X. Also f1(A) \ f1(B) = ;. Hence X is not g-connected. This is a contradiction. Therefore Y is connected. h Lemma 2.28 [2, Proposition 10]. For a topological space X, the following are equivalent: (i) X is g-connected. (ii) The only subset of X which are both g-open and g-closed are the empty set ; and X. Theorem 2.29. Let X be g-connected. Then each contra g-continuous maps of X into a discrete space Y with at least two points is a constant map.
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Proof. Let f : X ! Y be a contra g-continuous map. Then X is covered by g-open and g-closed covering {f1({y}) : y 2 Y}. By assumption f1({y}) = ; or X for each y 2 Y. If f1({y}) = ; for all y 2 Y then f fails to be a map. Then, there exists only one point y 2 Y such that f1({y}) 5 ; and hence f1({y}) = X which shows that f is a constant map. Recall that a function is called preclosed [12] if the image of every closed subset of X is preclosed in Y. A space X is called locally indiscrete [26] if every open set is closed. h Theorem 2.30. Let f : X ! Y be a surjective preclosed contra g-continuous function. If X is a g-space, then Y is locally indiscrete. Proof. Suppose that V is open in Y. By hypothesis f is contra g-continuous and therefore f1(V) = U is g-closed in X. Since X is a g-space, the set U is closed in X. Since f is preclosed, then V is also preclosed in Y. Now we have Cl(V) = Cl(Int(V)) V. This means that V is closed and hence Y is locally indiscrete. h We close with the following question: Question 2.31. Let X be g-connected and Y be T1. Is it true that if f is contra g-continuous, then f is constant?
References [1] Abd El-Monsef ME, El-Deeb SN, Mahmoud RA. b-open sets and b-continuous mappings. Bull Fac Assiut Univ 1983;12:77–90. [2] Balachandran K, Sundaram P, Maki H. On generalized continuous maps in topological spaces. Mem Fac Sci Kochi Univ (Math) 1991;12:5–13. [3] Caldas M. On g-closed sets and g-continuous mappings. Kyungpook Math J 1993;33:205–9. [4] Caldas M. Further results on generalized open mappings in topological spaces. Bull Cal Math Soc 1996;88:37–42. [5] Caldas M, Jafari S. On g-US spaces. Stud Cercet Stiint, Ser Mat, Univ Bacau 2005;14:13–20. [6] Caldas M, Jafari S. Some properties of contra-b-continuous functions. Mem Fac Sci Kochi Univ Ser A Math 2001;22:19–28. [7] Dontchev J. Contra-continuous functions and strongly S-closed spaces. Int J Math Math Sci 1996;19:303–10. [8] Dontchev J, Noiri T. Contra-semicontinuous functions. Math Pannonica 1999;10:159–68. [9] Dunham W, Levine N. Further results on generalized closed sets in topology. Kyungpook Math J 1980;20:169–75. [10] Dunham W. T1/2-spaces. Kyungpook Math J 1977;17:161–9. [11] Dunham W. A new closure operator for non-T1 topologies. Kyungpook Math J 1982;22:55–60. [12] El-Deeb SN, Hasanein IA, Mashhour AS, Noiri T. On p-regular spaces. Bull Math de la Soc Sci Math de la RS Roumanie 1983; 27:311–5. [13] El-Naschie MS. On the uncertainty of cantorian geometry and the two-slit experiment. Chaos, Solitons & Fractals 1998;9(3): 517–29. [14] El-Naschie MS. Quantum gravity from descriptive set theory. Chaos, Solitons & fractals 2004;19:1339–44. [15] El-Naschie MS. On the certification of heterotic strings, M theory and e1 theory. Chaos, Solitons & Fractals 2000:2397–408. [16] Jafari S, Noiri T. Contra-a-continuous functions between topological spaces. Iran Int J Sci 2001;2:153–67. [17] Jafari S, Noiri T. Contra-super-continuous functions. Ann Univ Sci Budapest 1999;42:27–34. [18] Jafari S, Noiri T. On contra-precontinuous functions. Bull Malaysian Math Sci Soc 2002;25(2):115–28. [19] Landi G. An introduction to noncommutative spaces and their geometris. Lecture notes in physics. New York: Springer-Verlag; 1997. [20] Levine N. Generalized closed sets in topology. Rend Circ Mat Palermo 1970;19(2):89–96. [21] Levine N. Semi-open sets and semi-continuity in topological spaces. Amer Math Monthly 1961;68:44–6. [22] Malghan SR. Generalized closed maps. J Karnatak Univ Sci 1982;27:82–8. [23] Mashhour AS, Abd El-Monsef ME, El-Deeb SN. On precontinuous and weak precontinuous mappings. Proc Math Phys Soc Egypt 1982;53:47–53. [24] Moore ELF, Peters TJ. Computational topology for geometric design and molecular design. In: Ferguson DR, Peters TJ, editors. Mathematics in industry-challenges and frontiers 2003. SIAM; 2005. [25] Mrsˇevic´ M. On pairwise R0 and pairwise R1 bitopological spaces. Bull Math Soc Sci Math RS Roumanie 1986;30(78):141–8. [26] Nieminen T. On ultrapseudocompact and related topics. Ann Acad Sci Fenn Ser A I Math 1977;3:185–205. [27] Nja˚stad O. On some classes of nearly open sets. Pacific J Math 1965;15:961–70. [28] Pawlak Z. Rough sets: theoretical aspects of reasoning about data. System theory, knowledge engineering and problem solving, vol. 9. Dordrecht: Kluwer; 1991. [29] Rosen DW, Peters TJ. The role of topology in engineering design research. Res Eng Des 1996;2:81–98. [30] Smyth MB. Topology. In: Abramsky S, Gabbay DM, Maibaum TSE, editors. Handbook of logic in computer sciences, vol. 1. Clarendon Press; 1992. p. 641–761. [31] Staum R. The algebra of bounded continuous functions into a nonarchimedean field. Pacific J Math 1974;50:169–85. [32] Witten W. Phys Today 1996:24–30.
A new generalization of contra-continuity via Levine’s g-closed sets Miguel Caldas
a,*
, Saeid Jafari b, Takashi Noiri c, Marilda Simo˜es
d
a
Departamento de Matematica Aplicada, Universidade Federal Fluminense, Rua Mario Santos Braga, s/n 24020-140, Niteroi, RJ, Brazil b College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark c Yatsushiro College of Technology, 2627 Hirayama shinmachi, Yatsushiro-shi, Kumamoto-ken 866-8501, Japan d Dipartimento Di Matematica ‘‘Guido Castelnuovo’’, Universita´ Di Roma ‘‘La Sapienza’’, 00185 Roma, Italy Accepted 13 December 2005
Abstract In [Dontchev J. Contra-continuous functions and strongly S-closed spaces. Int J Math Math Sci 1996;19:303–10], Dontchev introduced and investigated a new notion of continuity called contra-continuity. Recently, Jafari and Noiri [Jafari S, Noiri T. Contra-a-continuous functions between topological spaces. Iran Int J Sci 2001;2:153–67, Jafari S, Noiri T. Contra-super-continuous functions. Ann Univ Sci Budapest 1999;42:27–34, Jafari S, Noiri T. On contraprecontinuous functions. Bull Malaysian Math Sci Soc 2002;25(2):115–28] introduced new generalizations of contracontinuity called contra-a-continuity, contra-super-continuity and contra-precontinuity. In this paper, we introduce and investigate a generalization of contra-continuity by utilizing Levine’s generalized closed sets. 2006 Elsevier Ltd. All rights reserved.
1. Introduction and preliminaries General Topology has shown its fruitfulness in both the pure and applied directions. In data mining [28], computational topology for geometric design and molecular design [24], computer-aided geometric design and engineering design (briefly CAGD), digital topology, information systems, non-commutative geometry and its application to particle physics [19], one can observe the influence made in these realms of applied research by general topological spaces, properties and structures. Rosen and Peters [29] have used topology as a body of mathematics that could unify diverse areas of CAGD and engineering design research. They have presented several examples of the application of topology to CAGD and design. The concept of closedness is fundamental with respect to the investigation of general topological spaces. Levine [20] initiated the study of the so-called g-closed set and by doing this, he generalized the concept of closedness. The concept of g-closed sets were also considered by Dunham [11] in 1982 and by Dunham and Levine [9] in 1980. Balachandran et al. [2] in 1991, defined a new class of mappings called generalized continuous (briefly g-continuous) mappings which *
Corresponding author. E-mail addresses: [email protected]ff.br (M. Caldas), [email protected] (S. Jafari), [email protected] (T. Noiri), [email protected] (M. Simo˜es). 0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.12.032
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contains the class of continuous mappings. Since the advent of these notions, several research papers with interesting results in different respects came to existence (see, [2–4,9–11,22]). Recently, Jafari and Noiri introduced and investigated the notions of contra-precontinuity [18], contra-a-continuity [16] and contra-super-continuity [17] as a continuation of research done by Dontchev [7] and Dontchev and Noiri [8] on the interesting notions of contra-continuity and contrasemi-continuity, respectively. Caldas and Jafari [6] introduced and investigated the notion of contra-b-continuous functions in topological spaces as a new generalization of both contra-semi-continuity and contra-precontinuity. The present note has as its purpose to investigate some fundamental properties of contra g-continuous functions by using g-open sets. We believe that the notions introduced in this paper have applications in a broader sense. For example one can talk about g-manifolds in which everything open is changed by g-open sets such as Hausdorffness by g-T2 (defined as Hausdorff but by changing open with g-open), also continuity by contra g-continuity or g-continuity. We know that manifolds are fundamental in many branches of science in general and mathematical physics in special. On g-manifolds, one can impose GO-compactness [2] which its definition is similar to the definition of compactness by changing open with gopen. Then one can talk about Riemannian g-manifolds and semi-Riemannian g-manifolds. Indeed one can also talk about g-paracompact manifolds by using g-open sets instead of open sets in paracompact manifolds. The same is true with the definition of g-orbifold and lots of other geometrical objects. Moreover the notion of kernel of a set has applications in computer science [30]. This notion is used in Theorem 2.2 and Corollary 2.3 of this paper. It should be mentioned that the present work may become relevant to the work of Witten [32] and El-Naschie [13,14]. Also the fuzzy topological version of the notions and results introduced in this paper are very important since Professor El-Naschie has recently shown in [15] that the notion of fuzzy topology my be relevant to quantum particle physics in connection with string theory and e1 theory. Throughout this paper, (X, s) and (Y, r) (or X and Y) always denote topological spaces. A subset A of X is said to be regular open (resp. regular closed) if A = Int(Cl(A)) (resp. A = Cl(Int(A))) where Cl(A) and Int(A) denote the closure and interior of A, respectively. A subset A of a space X is called preopen [23] (resp. semi-open [21], a-open [27], b-open [1]) if A Int(Cl(A)) (resp. A Cl(Int(A)), A Int(Cl(Int(A))), A Cl(Int(Cl(A)))). The complement of a preopen (resp. semi-open, a-open, b-open) set is said to be preclosed (resp. semi-closed, a-closed, b-closed). The collection of all closed (resp. semi-open, clopen) subsets of X will be denoted by C(X) (resp. SO(X), CO(X)). We set C(X, x) = {V 2 C(X)/x 2 V} for x 2 X. We define similarly CO(X, x). A subset B of a topological space X is called g-closed in X [20] if Cl(B) G whenever B G and G is open in X. The union of two g-closed sets is a g-closed set. A subset A is called g-open in X if its complement, X A = Ac is g-closed. The intersection of all g-closed sets containing a set A is called the g-closure of A [11] and is denoted by Cl*(A). This is, for any A X, Cl*(A) = \{F X : A F and F is g-closed}. If A X, then A Cl*(A) Cl(A). The collection of all g-closed (resp. g-open) subsets of X will be denoted by GC(X) (resp. GO(X)). We set GC(X, x) = {V 2 GC(X)/x 2 V} for x 2 X. We define similarly GO(X, x). If A is g-open in X and B is g-open in Y, then A · B is g-open in X · Y [20]. Let p be a point of X and N be a subset of X. N is called a g-neighborhood of p in X [3] if there exists a g-open set O of X such that p 2 O N. Lemma 1.1 [3]. Let A be a subset of X. Then, p 2 Cl*(A) if and only if for any g-neighborhood Np of p in X, A \ Np 5 /. A mapping f : X ! Y from a topological space X into a topological space Y is called g-continuous [2] if the inverse image of every closed set in Y is g-closed in X. Observe that a mapping f : X ! Y is g-continuous if and only if the inverse image of every open set in Y is g-open in X.
2. Contra g-continuous functions Definition 1. A function f : X ! Y is said to be contra g-continuous if f1(V) is g-closed in X for each open set V of Y. Definition 2. Let A be a subset of a space (X, s). The set \{U 2 sjA U} is called the kernel of A [25] and is denoted by ker(A). Lemma 2.1 [17]. The following properties hold for subsets A, B of a space X: (1) x 2 ker(A) if and only if A \ F 5 ; for any F 2 C(X, x). (2) A ker(A) and A = ker(A) if A is open in X. (3) If A B, then ker(A) ker(B).
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Theorem 2.2. Let f : X ! Y be a function from a topological space X into a topological space Y. (i) The following statements are equivalent. (a) f is contra g-continuous. (b) The inverse image of each closed set in Y is g-open in X. (ii) The following statements are equivalent. (a) For each point x in X and each closed set V in Y with f(x) 2 V, there is a g-open set U in X containing x such that f(U) V. (b) For every subset A of X, f(Cl*(A)) ker(f(A)) holds. (c) For each subset B of Y, Cl*(f1(B)) f1(ker(B)). Proof (i) (a) M (b): see Definition 1. (ii) (a) ! (b): Let A be any subset of X. Suppose that y 62 ker(f(A)). Then, by Lemma 2.1 there exists V 2 C(Y, y) such that f(A) \ V = ;. For any x 2 f1(V), by (a) there exists Ux 2 GO(X, x) such that f(Ux) V. Hence f(A \ Ux) f(A) \ f(Ux) f(A) \ V = ; and A \ Ux = ;. This shows that x 62 Cl*(A) for any x 2 f1(V). Therefore, f1(V) \ Cl*(A) = ; and hence V \ f(Cl*(A)) = ;. Thus, y 62 f(Cl*(A)). Consequently, we obtain f(Cl*(A)) ker(f(A)). (b) ! (c): Let B be any subset of Y. By (b) and Lemma 2.1, we have f(Cl*(f1(B))) ker(ff1(B)) ker(B) and Cl*(f1(B)) f1(ker(B)). (c) ! (a): Suppose that (c) holds. Let x 2 X and V 2 C(Y, f(x)). By (c) and Lemma 2.1, Cl*(f1(YnV)) f1(ker(YnV)) = f1(YnV) and hence Cl*(f1(YnV)) = f1(YnV). Since x 2 Xnf1(V), there exists U 2 GO(X, x) such that U \ f1(YnV) = ;. Therefore, we obtain f(U) V. h Corollary 2.3. Assume that Cl*(A) is g-closed for each subset A of X. Then the following statements are equivalent for a function f : X ! Y: (1) (2) (3) (4) (5)
f is contra g-continuous, for every closed subset F of Y, f1(F) 2 GO(X), for each x 2 X and each F 2 C(Y, f(x)), there exists U 2 GO(X, x) such that f(U) F, f(Cl*(A)) ker(f(A)) for every subset A of X, Cl*(f1(B)) f1(ker(B)) for every subset B of Y.
Proof. The implications (1) ! (2) and (2) ! (3) are obvious. (3) ! (4) ! (5): Theorem 2.2(ii). (5) ! (1): Let V be any open set of Y. Then, by Lemma 2.1 we have Cl*(f1(V)) f1(ker(V)) = f1(V) and Cl*(f1(V)) = f1(V). This shows that f1(V) is g-closed in X [3, Remark 4]. h Remark 2.4. The following two examples show that g-continuity and contra g-continuity are independent concepts. Example 2.5. The identity function on the real line (with the usual topology) is continuous and hence g-continuous but not contra g-continuous, since the preimage of each singleton fails to be g-open. Example 2.6. Let X = {a, b} be the Sierpinski space with the topology s = {;, {a}, X}. Let f : X ! X be defined by f(a) = b and f(b) = a. Since the inverse image of every open set is g-closed, then f is contra g-continuous, but f1({a}) is not g-open in (X, s). Therefore f is not g-continuous. Definition 3. A function f : X ! Y is said to be contra-continuous [7] (resp. contra-a-continuous [16], contra-precontinuous [18], contra-semi-continuous [8], contra-b-continuous [6] if for each open set V of Y, f1(V) is closed (resp. a-closed, preclosed, semi-closed, b-closed) in X.
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For the functions defined above, we have the following implications: contra-continuity # contra g-continuity
! contra-a-continuity # contra-semi-continuity
! contra-precontinuity # ! contra-b-continuity
Remark 2.7. It should be mentioned that none of these implications is reversible as shown by the examples stated below. Example 2.8 [16]. Let X = {a, b, c}, s = {;, {a}, X} and r = {;, {b}, {c}, {b, c}, X}. Then the identity function f : (X, s) ! (X, r) is contra-a-continuous but not contra-continuous. Example 2.9. Let X = {a, b, c}, s = {;, {a}, X} and r = {;, {b}, {c}, {b, c}, X}. Then the identity function f : (X, s) ! (X, r) is contra g-continuous but not contra-continuous. Example 2.10 [8]. A contra-semi-continuous function need not be contra-precontinuous. Let f : R ! R be the function f(x) = [x], where [x] is the Gaussian symbol. If V is a closed subset of the real line, its preimage U = f1(V) is the union of the intervals of the form [n, n + 1], n 2 Z; hence U is semi-open being union of semi-open sets. But f is not contraprecontinuous, since f1(0.5, 1.5) = [1, 2) is not preclosed in R. Example 2.11 [8]. A contra-precontinuous function need not be contra-semi-continuous. Let X = {a, b}, s = {;, X} and r = {;, {a}, X}. The identity function f : (X, s) ! (Y, r) is contra-precontinuous as only the trivial subsets of X are open in (X, s). However, f1({a}) = {a} is not semi-closed in (X, s); hence f is not contra-semi-continuous. Theorem 2.12. Assume that Cl*(A) is g-closed for each subset A of X. If a function f : X ! Y is contra g-continuous and Y is regular, then f is g-continuous. Proof. Let x be an arbitrary point of X and V an open set of Y containing f(x). Since Y is regular, there exists an open set W in Y containing f(x) such that Cl(W) V. Since f is contra g-continuous, so by Corollary 2.3, there exists U 2 GO(X, x) such that f(U) Cl(W). Then f(U) Cl(W) V. Hence, f is g-continuous. h Definition 4. A space (X, s) is said to be: (1) locally g-indiscrete if every g-open set of X is closed in X, (2) g-space if every g-open set of X is open in X, (3) gS-space if and only if every g-open subset of X is semi-open. The following theorem follows immediately Definition 4: Theorem 2.13. If a function f : X ! Y is contra g-continuous and X is a gS-space (resp. g-space, locally g-indiscrete), then f is contra-semi-continuous (resp. contra-continuous, continuous). Recall that a topological space X is said to be (1) g-T2 [5] if for each pair of distinct points x and y in X there exist U 2 GO(X, x) and V 2 GO(X, y) such that U \ V = ;. (2) Ultra Hausdorff [31] if for each pair of distinct points x and y in X there exist U 2 CO(X, x) and V 2 CO(X, y) such that U \ V = ;. Observe that in the following theorem and its two corollaries, we will assume that Cl*(A) is g-closed for every subset A of a topological space. Theorem 2.14. If X is a topological space and for each pair of distinct points x1 and x2 in X there exists a function f of X into a Urysohn space Y such that f(x1) 5 f(x2) and f is contra g-continuous at x1 and x2, then X is g-T2.
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Proof. Let x1 and x2 be any distinct points in X. Then by hypothesis, there is a Urysohn space Y and a function f : X ! Y, which satisfies the conditions of the theorem. Let yi = f(xi) for i = 1, 2. Then y1 5 y2. Since Y is Urysohn, there exist open neighbourhoods V y 1 and V y 2 of y1 and y2 respectively in Y such that ClðV y 1 Þ \ ClðV y 2 Þ ¼ ;. Since f is contra g-continuous at xi, there exists a g-open neighbourhood U xi of xi in X such that f ðU xi Þ ClðV y i Þ for i = 1, 2. Hence we get U x1 \ U x2 ¼ ; because ClðV y 1 Þ \ ClðV y 2 Þ ¼ ;. Then X is g-T2. h Corollary 2.15. If f is a contra g-continuous injection of a topological space X into a Urysohn space Y, then X is g-T2. Proof. For each pair of distinct points x1 and x2 in X, f is a contra g-continuous function of X into a Urysohn space Y such that f(x1) 5 f(x2) because f is injective. Hence by Theorem 2.14, X is g-T2. h Corollary 2.16. If f is a contra g-continuous injection of a topological space X into a Ultra Hausdorff space Y, then X is g-T2. Proof. Let x1 and x2 be any distinct points in X. Then since f is injective and Y is Ultra Hausdorff, f(x1) 5 f(x2) and there exist V1, V2 2 CO(Y) such that f(x1) 2 V1, f(x2) 2 V2 and V1 \ V2 = ;. Then xi 2 f1(Vi) 2 GO(X) for i = 1, 2 and f1(V1) \ f1(V2) = ;. Thus X is g-T2. h Lemma 2.17 [20]. If Ai is a g-open set in a topological space Xi, for i = 1, 2, . . . , n, then A1 · · An is also g-open in the product space X1 · · Xn. Theorem 2.18. Let f1 : X1 ! Y and f2 : X2 ! Y be two functions, where (1) GO(X1 · X2) is closed under ordinary unions, (2) Y is a Urysohn space, (3) f1 and f2 are contra g-continuous. Then {(x1, x2)/f1(x1) = f2(x2)} is g-closed in the product space X1 · X2. Proof. Let A denote the set {(x1, x2)/f1(x1) = f2(x2)}. In order to show that A is g-closed, we show that (X1 · X2)nA is gopen. Let (x1, x2) 62 A. Then f1(x1) 5 f2(x2) . Since Y is Urysohn, there exist open sets V1 and V2 containing f1(x1) and f2(x2), respectively such that Cl(V1) \ Cl(V2) = ;. Since fi (i = 1, 2) is contra g-continuous, fi1 ðClðV i ÞÞ is a g-open set containing xi in Xi(i = 1, 2). Hence by Lemma 2.17 f11 ðClðV 1 ÞÞ f21 ðClðV 2 ÞÞ is g-open. Further ðx1 ; x2 Þ 2 f11 ðClðV 1 ÞÞ f21 ðClðV 2 ÞÞ ðX 1 X 2 Þ n A. It follows that (X1 · X2)nA is g-open. Thus A is g-closed in the product space X1 · X2. h Corollary 2.19. Let GO(X · X) be closed under unions. If f : X ! Y is contra g-continuous and Y is a Urysohn space, then A = {(x1, x2)/f(x1) = f(x2)} is g-closed in the product space X · X. Definition 5. A topological space X is said to be (1) g-normal if each pair of non-empty disjoint closed sets can be separated by disjoint g-open sets. (2) Ultra normal [31] if each pair of non-empty disjoint closed sets can be separated by disjoint clopen sets. Theorem 2.20. If f : X ! Y is a contra g-continuous, closed injection and Y is ultra normal, then X is g-normal. Proof. Let F1 and F2 be disjoint closed subsets of X. Since f is closed and injective, f(F1) and f(F2) are disjoint closed subsets of Y. Since Y is ultra normal, f(F1) and f(F2) are separated by disjoint clopen sets V1 and V2, respectively. Hence Fi f1(Vi), f1(Vi) 2 GO(X) for i = 1, 2 and f1(V1) \ f1(V2) = ;. Thus X is g-normal. h Theorem 2.21. Let GO(X) be closed under unions. Let f : X ! Y be a function and g : X ! X · Y the graph function, given by g(x) = (x, f(x)) for every x 2 X. Then f is contra g-continuous if and only if g is contra g-continuous. Proof. Let x 2 X and W be a closed subset of X · Y containing g(x). Then W \ ({x} · Y) is closed in {x} · Y containing g(x). Also {x} · Y is homeomorphic to Y. Hence {y 2 Y : (x, y) 2 W} is a closed subset of Y. Since f is contra
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g-continuous, [{f1(y) : (x, y) 2 W} is a g-open subset of X . Further x 2 [{f1(y): (x, y) 2 W} g1(W). Hence g1(W) is g-open. Then g is contra g-continuous. Conversely, let F be a closed subset of Y. Then X · F is a closed subset of X · Y. Since g is contra g-continuous, g1(X · F) is a g-open subset of X. Also g1(X · F) = f1(F). Hence f is contra g-continuous. h Theorem 2.22. If f : X ! Y is a contra g-continuous function and g : Y ! Z is a continuous function, then g f : X ! Z is contra g-continuous. Recall that, a topological space X is said to be T1/2-space [20] if every g-closed set is closed in X. Theorem 2.23. Let X and Z be any topological spaces and Y be a T1/2-space. If f : X ! Y is contra g-continuous and g : Y ! Z is g-continuous, then g f : X ! Z is contra g-continuous. Proof. It follows from definitions.
h
Q g-continuous function. Theorem 2.24. Let {Xk : k 2 X} be any family of topological spaces. If f : X ! X k is a contra Q Then Prk f : X ! Xk is contra g-continuous for each k 2 X, where Prk is the projection of X k onto Xk. Q X k . Since Proof. We shall consider a fixed k 2 X. Suppose Uk is an arbitrary open set in Xk. Then Pr1 k ðU k Þ is open in 1 f is contra g-continuous, we have by definition f 1 ðPr1 k ðU k ÞÞ ¼ ðPrk f Þ ðU k Þ is g-closed in X. Therefore Prk f is contra g-continuous. h Recall that for a function f : X ! Y, the subset {(x, f(x)) : x 2 X} X · Y is called the graph of f and is denoted by G(f). Definition 6. A graph G(f) of a function f : X ! Y is said to be contra g-closed if for each (x, y) 2 (X · Y)nG(f), there exist U 2 GO(X) containing x and V 2 C(Y) containing y such that (U · V) \ G(f) = ;. Lemma 2.25. A graph G(f) of a function f : X ! Y is contra g-closed in X · Y if and only if for each (x, y) 2 (X · Y)nG(f), there exist U 2 GO(X) containing x and V 2 C(Y) containing y such that f(U) \ V = ;. Theorem 2.26. If f : X ! Y is contra g-continuous and Y is Urysohn, then G(f) is contra g-closed in X · Y. Proof. Let (x, y) 2 (X · Y)nG(f), then f(x) 5 y and there exist open sets V, W such that f(x) 2 V, y 2 W and Cl(V) \ Cl(W) = ;. Since f is contra g-continuous, there exists U 2 GO(X, x) such that f(U) Cl(V). Therefore, we obtain f(U) \ Cl(W) = ;. This shows that G(f) is contra g-closed in X · Y. h Definition 7. A topological space X is said to be g-connected [2] if X cannot be expressed as the union of two disjoint non-empty g-open subsets of X. Theorem 2.27. A contra g-continuous image of a g-connected space is connected. Proof. Let f : X ! Y be a contra g-continuous function of a g-connected space X onto a topological space Y. If possible, let Y be disconnected. Let A and B form a disconnection of Y. Then A and B are clopen and Y = A [ B where A \ B = ;. Since f is a contra g-continuous function, X = f1(Y) = f1(A [ B) = f1(A) [ f1(B), where f1(A) and f1(B) are non-empty g-open sets in X. Also f1(A) \ f1(B) = ;. Hence X is not g-connected. This is a contradiction. Therefore Y is connected. h Lemma 2.28 [2, Proposition 10]. For a topological space X, the following are equivalent: (i) X is g-connected. (ii) The only subset of X which are both g-open and g-closed are the empty set ; and X. Theorem 2.29. Let X be g-connected. Then each contra g-continuous maps of X into a discrete space Y with at least two points is a constant map.
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Proof. Let f : X ! Y be a contra g-continuous map. Then X is covered by g-open and g-closed covering {f1({y}) : y 2 Y}. By assumption f1({y}) = ; or X for each y 2 Y. If f1({y}) = ; for all y 2 Y then f fails to be a map. Then, there exists only one point y 2 Y such that f1({y}) 5 ; and hence f1({y}) = X which shows that f is a constant map. Recall that a function is called preclosed [12] if the image of every closed subset of X is preclosed in Y. A space X is called locally indiscrete [26] if every open set is closed. h Theorem 2.30. Let f : X ! Y be a surjective preclosed contra g-continuous function. If X is a g-space, then Y is locally indiscrete. Proof. Suppose that V is open in Y. By hypothesis f is contra g-continuous and therefore f1(V) = U is g-closed in X. Since X is a g-space, the set U is closed in X. Since f is preclosed, then V is also preclosed in Y. Now we have Cl(V) = Cl(Int(V)) V. This means that V is closed and hence Y is locally indiscrete. h We close with the following question: Question 2.31. Let X be g-connected and Y be T1. Is it true that if f is contra g-continuous, then f is constant?
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