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King Fahd University of Petroleum & Minerals DEPARTMENT OF MATHEMATICAL SCIENCES Technical Report Series TR 352 May 2006 Contra-Gamma-Continuous Mappings in Topological Spaces Raja Mohammad Latif DHAHRAN 31261 ● SAUDI ARABIA ● www.kfupm.edu.sa/math/ ● E-mail: [email protected] Contra-Gamma-Continuous Mappings in Topological Spacs Raja Mohammad Latif Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran 31261 Saudi Arabia [email protected] Abstract The notion of semi-convergence of …lters was introduced by Latif (1999) who investigated some characterizations related to semi-open continuous functions. In the spirit of Latif (1999) ; Min (2002) used the idea of semi-convergence of …lters to introduce a new class of sets, called closure, interior and open sets, and the notions of continuity and investigated some properties. In this paper, we apply the notion of open sets in topological spaces to present and study certain properties and characterizations of contra as a new generalization of contra continuity continuity [Dontchev; 1996] : ————————————————————————————— 2000 Mathematics Subject Classi…cation: 54A05; 54C08: Key Words and Phrases: Topological Space, closed; compact; strongly S open set, closed, contra 1 closed set, contra continuity: 1. Introduction The notion of open set (originally called sets) in topological spaces was introduced by Min [M in; 2002]. For these sets, in [Latif; (T R#331) 2005] we introduced the notions of derived, border; f rontier, and exterior of a set and showed that some of their properties are analogous to those for open sets. Also, we gave some additional properties of closure and interior of a set. We also continued to explore further properties and characterizations of irresolute and open mappings. In [Latif; (T R#332) 2005] we also intro- duced and studied properties and characterizations of pre continuous; closed; pre open and closed mappings. In 1996; Dontchev [Dontchev; 1996] introduced a new class of functions called contra-continuous functions. Recently, Dontchev and Noiri [1999] introduced and studied, among others, a new weaker form of this class of functions called contrasemicontinuous functions. They also introduced the notion of RC-continuity [Dontchev and N oiri; 1999] which is weaker than contra-continuity and stronger than continuity [T ong; 1998] : Jafari and Noiri [Jaf ari and N oiri; 1999] introduced and studied a new class of functions called contra-supper-continuous functions which lies between classes of RC-continuous functions and contra-continuous functions. This paper is devoted to introduce and investigate a new class of functions called contra- -continuous functions which is weaker than contra-continuous functions. 2. Preliminaries Throughout this paper, (X; ) (simply X) always mean topological space on which no separation axioms are assumed unless explicitly stated. Let S be a subset of X: The closure (resp., interior) of S will be denoted by Cl (S) (resp., Int (S)): A subset S of X is called a semi-open set [Levine; 1963] (resp., 1965]) if S Cl [Int (S)] ( resp:; S semi-open set (resp., open set [N jastad; Int [Cl (Int (S))]): The complement of a open set) is called semi-closed set (resp., 2 closed set). The family of all semi-open sets (resp., will be denoted by SO (X) (resp., open sets) in a topological space (X; ) ). The complement of a semi-opem set S is called a semi-closed set. The intersection of all semi-closed sets containing A is called the semi-closure of A; denoted by Cls (A) : A subset M (x) of a space X is called a semi-neighbourhood of a point x 2 X if there exists a semi-open set S such that x 2 S M (x) : In [Latif; 1999] Latif introduced the notion of semi- convergence of …lters and investigated some characterizations related to semi-open continuous function. Now, we recall the concept of semi-convergence of …lters. Let S (x) = fA 2 SO (X) : x 2 Ag and let Sx = fA is f inite and \ that For any …lter X : there exists S (x) such Ag: Then, Sx is called the semi-neighbourhood …lter at x: on X, we say that semi-converges to x if and only if is …ner than the semi-neighbourhood …lter at x: A subset U of X is called a to x and x 2 U; U 2 open set if whenever a …lter : The complement of a set. The intersection of all We denote the family of all that open set is called a closed sets containing A is called the A; denoted by Cl (A) : A subset A is also semi-converges closed closure of closed if and only if A = Cl (A) : open sets of (X; ) by : It is shown in [M in; 2002] is a topology on X: In a topological space (X; ); it is always true that SO (X) : 3. Contra -Gamma-Continuous Mappings The purpose of this section is to explore properties and characterizations of contra continuous mappings. De…nition 3.1. A function f : (X; ) ! (Y; ) is called contra f 1 (V ) is continuous if closed in X for each open set V of Y: De…nition 3.2. [Dontchev; 1996] : A function f : (X; ) ! (Y; ) is called contra continuous if f 1 (V ) is closed in X for each open set V of Y: 3 De…nition 3.3. [Dontchev & N oiri; 1999] : A function f : (X; ) called contra 1 semicontinuous if f ! (Y; ) is (V ) is semi-closed in X for each open set V of Y: Remark 3.4. Every contra continuous function is contra continuous but not conversely as the following example shows. Example 3.5. Let X = fa; b; cg ; = fX; ; fagg and = fX; ; fbg ; fcg ; fb; cgg: Then the identity function f : (X; ) ! (X; ) is contra not contra continuous but continuous. De…nition 3.6. [M rsevic; 1986] : Let A be a subset of a space (X; ) : The set \ fU 2 U g is called the kernel of A and is denoted by Ker (A) : :A Lemma 3.7. The following properties hold for subsets A; B of a space X : (1) x 2 Ker (A) if and only if A \ F 6= for any closed subset F of X such that x 2 F: (2) A Ker (A) and A = Ker (A) if A is open in X: (3) A B; then Ker (A) Ker (B) : Theorem 3.8. The following are equivalent for a function f : (X; ) ! (Y; ) : (1) f is contra continuous; (2) for every closed subset F of Y; f 1 (F ) 2 ; (3) for each x 2 X and each closed subset F of Y containing f (x) ; there exists U2 such that x 2 U and f (U ) (4) f [Cl (A)] (5) Cl [f 1 F; Ker [f (A)] for every subset A of X; (B)] f 1 [Ker (B)] for every subset B of Y: Proof The implications (1) () (2) and (2) =) (3) are obvious. 4 1 (3) ) (2) : Let F be any closed subset of Y and x 2 f and there exists Ux 2 f 1 such that x 2 Ux and f (Ux ) 1 (F ) = [ fUx jx 2 f (F )g 2 (F ) : Then f (x) 2 F F: Therefore, we obtain : (2) =) (4) : Let A be any subset of X: Suppose that y 2 = Ker [f (A)] : Then by Lemma 3:7 there exists F a closed subset of Y such that y 2 F and f (A) \ 1 F = : Thus, we have A \ f obtain f [Cl (A)] \ F = (F ) = and Cl (A) \ f 1 (F ) = : Therefore, we and y 2 = f [Cl (f (A))] : This implies that f [Cl (A)] Ker [f (A)] : (4) =) (5) : Let B be any subset of Y . By (4) and Lemma 3:7 we have f [Cl (f Ker (B) and Cl [f 1 (B)] f 1 1 (V ) is 1 (V )] f 1 [Ker (V )] = f 1 (V ). This closed in X: Theorem 3.9. A function f : (X; ) ! (Y; ) is contra only if f : (X; (B))] [Ker (B)] : (5) =) (1) : Let V be any open set of Y: Then, by Lemma 3:7 we have Cl [f shows that f 1 ) ! (Y; ) is contra continuous if and continuous: De…nition 3.10. A subset A of a topological space is called a set if it is the intersection of open sets. Theorem 3.11. A function f : (X; ) ! (Y; ) is contra only if the inverse images of continuous if and sets are closed. De…nition 3.12 A function f : (X; ) ! (Y; ) is said to be ( ; s) open if f (U ) 2 SO (Y ) for every U 2 : Theorem 3.13. Prove that a mapping f : (X; ) ! (Y; ) is ( ; s) only if for each x 2 X; and U 2 set V open if and such that x 2 U; there exists a semi Y containing f (x) such that V Proof. Follows directly from De…nition 3:12: 5 f (U ) : open Theorem 3.14. Let f : (X; ) ! (Y; ) be ( ; s) a closed set containing f containing W such that f Proof. Let H = Y 1 1 (H) (X 1 [Cls (Ints (Cls (B)))] Proof. Cl [f 1 (B)] is exists semi f 1 closed set H 1 (W ) F; we have f (X ! (Y; ) be ( ; s) Cl [f 1 1 F) (Y 1 (H) = X f W): [f (X F )] closed B 1 Y such that f H f open and let B 1 Y: Then (B)] : closed in X containing f [Cls (Ints (Cls (B)))] (B) : By Theorem 3:14, there 1 (H) [Cls (Ints (Cls (H)))] Cl [f f 1 1 (H) (B)] : Thus, Cl [f Theorem 3.16. Prove that a function f : (X; ) ! (Y; ) is ( ; s) only if f [Int (A)] Ints [f (A)] ; for all A Proof. Necessity. Let A x 2 Ux Y F ) = F: Corollary 3.15. Let f : (X; ) f X is F: Since f is ( ; s) open; then H is semi closed and f X Y and F (W ) ; there exists a semi F ) : Since f f (X open. If W X: Ints [f (A)] : : Then by hypothesis, f [Int (U )] Int (U ) = U as U is open: Also Ints [f (U )] Ints [f (U )] : Thus f (U ) is semi Ints [f (U )] : Since f (U ) : Hence f (U ) = open in Y: So f is ( ; s) open: Theorem 3.17. Prove that a function f : (X; ) ! (Y; ) is ( ; s) only if Int [f 1 (B)] Proof. Necessity. Let B open; f [Int (f 1 f 1 [Ints (B)] ; for all B Y: Since Int [f (B))] is semi such that f (A) and by hypothesis, f (Ux ) 2 SO (Y ) : Hence f (x) 2 Ints [f (A)] : Thus f [Int (A)] Su¢ ciency. Let U 2 (B)] : open if and X: Let x 2 Int (A) : Then there exists Ux 2 A: So f (x) 2 f (Ux ) 1 1 (B)] is open if and Y: open in X and f is ( ; s) open in Y: Also we have f [Int (f 6 1 (B))] f [f f 1 1 1 B: Hence, f [Int (f (B)] Ints (B) : Therefore Int [f (B))] 1 (B)] [Ints (B)] : Su¢ ciency.Let A X: Then f (A) 1 (f (A))] 1 f Y: Hence by hypothesis, we obtain [Ints (f (A))] : Thus f [Int (A)] Int (A) Int [f for all A X: Hence, by Theorem 3:16; f is ( ; s) Ints [f (A)] ; open: We remark that the equality does not hold in the preceding two theorems as the following example shows. Example 3.18. Let X = Y = f1; 2g : Suppose and be the antidiscrete topology on X be the discrete topology on Y: Then A = f1g : Then = Int [f 1 and SO (Y ) = : Let f = Id:; = = f [Int (A)] 6= Ints [f (A)] = f1g : Let B = f1g (B)] 6= f 1 Y: Then [Ints (B)] = f1g : Theorem 3.19. Let f : (X; ) ! (Y; ) be a mapping. Then a necessary and su¢ cient condition for f to be ( ; s) open is that f 1 [Cls (B)] Cl [f 1 (B)] for every subset B of Y: Proof. Necessity. Assume f is ( ; s) f (x) 2 Cls (B) : Let U 2 is a semi Su¢ ciency. Let B 1 (Y Hence X 1 Y: Let x 2 f [Cls (B)] : Then such that x 2 U: Since f is ( ; s) open; then f (U ) open set in Y: Therefore, B \ f (U ) 6= : Then U \ f Hence x 2 Cl [f Cl [f open: Let B 1 (B)] : We conclude that f Y: Then (Y f 1 1 10 [Latif; 1993] ; Int [f follows that f is ( ; s) (B)] (B)] f [Cls (B)] Cl [f Y: By hypothesis, f B) B)] : This implies X Cl [f Cl [X 1 1 [Y f 1 1 (Y Cls (Y B)] X f 1 1 (B) 6= : 1 (B)] : [Cls (Y 1 [Cls (Y B)] B)] : B)] : By applying Theorem [Ints (B)] : Now from Theorem 3:17; it open: De…nition 3.20. A …lter base is said to be 7 convergent (resp: c convergent) to a point x in X if for any U 2 such that x 2 U (resp: closed set U such that x 2 U ); there exists B 2 such that B X U: Theorem 3.21. A function f : (X; ) ! (Y; ) is contra continuous if and only if for each point x 2 X and each …lter base converging to x; the …lter base f ( ) is c convergent to f (x) : Proof. Necessity. Let x 2 X and to x: Since f is contra be any …lter base in X such that converges continuous; then for any closed subset V of Y such that f (x) 2 V; there exists U 2 such that x 2 U and f (U ) converging to x; there exists a B 2 f (B) in X such that B V and therefore the …lter base f ( ) is c V: Since is U: This means that convergent to f (x) : Su¢ ciency. Let x 2 X and V be a closed subset of Y such that f (x) 2 V: Let = fU X:U 2 and x 2 U g : Then to x: Thus, there exists U 2 is a …lter base which such that f (U ) converges V: 4. Contra-Gamma-Closed Graph In this section we de…ne contra closed graph and study some of its charac- teristics. De…nition 4.1. The graph G (f ) = f(x; f (x)) jx 2 Xg of a function f : (X; ) ! (Y; ) is said to be contra there exists U 2 and we have (U closed if for each (x; y) 2 (X Y) such that x 2 U and V a closed subset of Y such that y 2 V V ) \ G (f ) = : Lemma 4.2. The graph G (f ) of a function f : (X; ) ! (Y; ) is contra in X G (f ) ; Y if and only if for each (x; y) 2 (X Y) closed G (f ) ; there exists U 2 such that x 2 U and V a closed subset of Y such that y 2 V and we have f (U ) \ V = : 8 Theorem 4.3. If f : (X; ) ! (Y; ) is contra then G (f ) is contra Proof. Let (x; y) 2 (X closed in X continuous and Y is Urysohn, Y: G (f ) : Then y 6= f (x) and there exist open sets Y) V , W such that f (x) 2 V; y 2 W and Cl (V ) \ Cl (W ) = contra continuous, there exists U 2 : Since f is such that x 2 U and f (U ) Cl (V ) : Therefore, we obtain f (U ) \ Cl (W ) = : This shows that G (f ) is contra closed: De…nition 4.4. [M in; 2002] : A function f : (X; ) continuous if f 1 (V ) 2 for every V 2 : Theorem 4.5. If f : (X; ) ! (Y; ) is contra closed in X Proof. Let (x; y) 2 (X ! (Y; ) is said to be continuous and Y is T1 ; then G (f ) is Y: G (f ) : Then f (x) 6= y and there exists an open Y) set V of Y such that f (x) 2 V and y 2 = V: Since f is exists a f (U )\(Y open neighbourhood of x such that f (U ) V)= and (Y continuous, there V: Therefore, we obtain V ) is a closed subset of Y such that y 2 (Y This shows that G (f ) is contra closed in X V ): Y: De…nition 4.6. [Dontchev; 1996] : A space X is said to be strongly S closed if every closed cover of X has a …nite subcover. A subset A of a space X is said to be strongly S closed if the subspace A is strongly S Theorem 4.7. If f : (X; ) ! (Y; ) has a contra inverse image of a strongly S closed set K of Y is Proof. Assume that K is a strongly S closed: closed graph, then the closed in X: closed set of Y and x 2 =f k 2 K; (x; k) 2 = G (f ) : By Lemma 4:2; there exist Uk a 1 (K) : For each open neighbourhood of x and Vk a closed subset of Y with k 2 Vk such that f (Uk ) \ Vk = : Since 9 fK \ Vk jk 2 Kg is a closed cover of the subspace K; there exists a …nite subset K1 K such that K [ fVk jk 2 K1 g : Then U = \ fUk jk 2 K1 g is a neighbourhood of x and f (U ) \ K = : Therefore U \ f x2 = Cl [f 1 (K)] : This shows that f Theorem 4.8. Let Y be a strongly S (Y; ) has a contra 1 (K) is (K) = and hence closed in X: closed space. If a function f : (X; ) ! closed graph, then f is contra Proof. Suppose that Y is strongly S continuous: closed and G (f ) is contra we show that an open set of Y is strongly S Y and fH : 1 open closed: First, closed: Let V be an open set of 2 rg be a cover of V by closed sets H of V: For each 2 r; there exists a closed set K of X such that H = K \ V: Then, the family fK j 2 rg [ f(Y V )g is a closed cover of Y: Since Y is strongly S there exists a …nite subset r0 closed , r such that Y = ([ fK j 2 r0 g) [ (Y V ): Therefore we obtain V = [ fK j 2 r0 g : This shows that V is strongly S closed: For any open set V; by Theorem 4:7 f contra 1 (V ) is closed in X and f is continuous: 5. Covering Properties In this section we study the properties of compact and strongly S under the contra closed spaces continuous functions. Theorem 5.1. Let f : (X; ) ! (Y; ) be a contra K be a compact subset of (X; continuous function. Let ) : Then f (K) is strongly S closed in Y: Proof. Let fH j 2 rg be any cover of f (K) by closed sets of the subspace f (K) : For each 2 r; there exists a closed set K of Y such that H = K \ f (K) : For each x 2 K; there exists 3:8; there exists Ux a (x) 2 r such that f (x) 2 K (x) and by Theorem open neighbourhood of x such that f (Ux ) Since the family fUx jx 2 Kg is a cover of K by 10 K (x) : open sets of X; there exists a …nite subset K0 of K such that K obtain f (K) f (K) = [ H [ fUx jx 2 K0 g : Therefore we [ ff (Ux ) jx 2 K0 g which is a subset of [ K (x) jx 2 K0 and hence f (K) is strongly S Corollary 5.2. If f : (X; ) ! (Y; ) is a contra (X; ) is compact, then Y is strongly S Theorem 5.3. If f : (X; ) (X; ) is a strongly S 2 K0 : Thus, closed: continuous surjection and closed: ! (Y; ) is contra continuous surjection and closed space, then (Y; ) is compact. Proof. Let fV j 2 rg be any open cover of Y: Then ff of X: Since f is contra 1 continuous; f 2 r: This implies that ff S (x) jx 1 1 (V ) j 2 rg is a cover (V ) is (V ) j 2 rg is a closed space X: We have X = [ ff 1 closed in X for each closed cover of the strongly (V ) j 2 r0 g for some …nite r0 of r: Since f is surjective, Y = [ fV j 2 r0 g : This shows that (Y; ) is compact. 6. Connected Spaces In this section we study the properties of connected spaces under the contra continuous functions. Theorem 6.1. If f : (X; ) (X; ! (Y; ) is contra continuous surjection and ) is connected, then (Y; ) is connected. Proof. Suppose Y is not connected. There exist nonempty disjoint open sets V1 and V2 such that Y = V1 [ V2 : Therefore, V1 and V2 are clopen in Y: Since f is contra continuous; f in X and hence disjoint and X = f 1 (V1 ) and f 1 clopen in X: Moreover, f 1 (V1 ) [ f 1 (V2 ) are 1 (V1 ) and f (V2 ) : This shows that (X; closed and 1 open (V2 ) are nonempty ) is not connected. De…nition 6.2. [Steen and Seebach; 1970] : A topological space (X; ) is said to be hyperconnected if the closure of every nonempty open set is the entire set X: It is well-known that every hyperconnected space is connected but not conversely. 11 Remark 6.3. In example 3:5; (X; ) is hyperconnected and f : (X; ) ! (X; ) is a continuous surjection, but (X; ) is not hyperconnected. This shows contra that contra continuous surjections do not preserve hyperconnectedness. De…nition 6.4. [Levine; 1961] : A function f : (X; ) ! (Y; ) is said to be weakly continuous if for each point x 2 X and each open set V of Y containing f (x) ; there exists an open set U containing x such that f (U ) Cl (V ) : Theorem 6.5. [N oiri; 1974] : If f : (X; ) ! (Y; ) is a weakly continuous surjection and X is connected, then Y is connected. Remark 6.6. Contra continuity and weak continuity are independent of each other. In Example 3:5; the function f is contra continuous but not weakly continuous. The following example shows that not every weakly continuous function is contra continuous: Example 6.7. Let X = fa; b; c; dg and = f ; X; fbg ; fcg ; fb; cg ; fa; bg ; fa; b; cg ; fb; c; dgg: De…ne a function f : (X; ) ! (X; ) as follows: f (a) = c; f (b) = d; f (c) = b and f (d) = a: Then f is weakly continuous [N eubrunnova; 1980] : However, f is not contra and f 1 continuous since fag is a closed set of (X; ) (fag) = fdg is not open in (X; ) : Acknowledgement The author is highly and gratefully indebted to the King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, for providing necessary research facilities during the preparation of this paper. References 1. J. Dontchev, Contra continuous f unctions and strongly S Internat. J. Math. Sci., 19 (1996) ; 303 12 310: closed spaces, 2. J. Dontchev, and T. Noiri, Contra nonica, 10 (1999) ; 159 semicontinuous f unctions; Math. Pan- 168: 3. S. Jafari, and T. Noiri, Contra Sci. 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