Department of Mathematics, University of Sargodha, Mandi Bahauddin Campus, Pakistan

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1 208 IJSRST Volume 4 Issue 9 Print ISSN : Online ISSN : X Themed Section: Science and Technology Contra Bc Continuous Functions in Topological Spaces Raja Mohammad Latif *, Muhammad Rafiq Raja 2, Muhammad Razaq 3 * Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al-Khobar, Kingdom of Saudi Arabia 2 Department of Mathematics, University of Sargodha, Mandi Bahauddin Campus, Pakistan 3 Department of Mathematics, Horizon Degree College and Commerce College, Bhaun Road, Chakwal, Pakistan ABSTRACT In this paper by means of Bc open sets, we introduce and investigate certain ramifications of contra continuous and allied functions, namely, contra Bc continuous, almost Bc continuous, almost weakly Bc continuous and almost contra Bc continuous functions along with their several properties, characterizations and natural relationships Further, we introduce new types of graphs, called Bc closed, contra Bc closed and strongly contra Bc closed graphs via Bc open sets Several characterizations and properties of such notions are investigated Keywords : Almost contra Bc continuous, contra Bc closed Graph, contra Bc continuous, Bc connected, Bc open 200 Mathematics Subject Classification: 54A05, 54A0, 54C05 I INTRODUCTION In recent literature, we find many topologists have focused their research in the direction of investigating different types of generalized continuity One of the outcomes of their research leads to the initiation of different orientations of contra continuous functions The notion of contra continuity was first investigated by Dontchev [7] Subsequently, Jafari and Noiri [5, 6] ehibited contra continuous and contra pre continuous functions Contra precontinuous [0] was studied by Ekici and Noiri contra e continuous functions [2] was studied by Gosh and Basu A good number of researchers have also initiated different types of contra continuous like functions, some of which are found in the papers [6, 8,, 8, 23] In [] Al Abdullah and Abed introduced a new class of sets in a topological space, known as Bc open sets In this paper, we employ this notion of Bc open sets to introduce and investigate contra continuous functions, called contra Bc continuous functions We study fundamental properties of contra Bc continuous functions and use such functions to characterize Bc connectedness We introduce and study the properties of almost contra Bc continuous functions as well We also introduce and study the notions of Bc closed, contra Bc closed and slightly contra closed graphs II PRELIMINARIES In this paper, spaces X and Y always represent topological spaces X, and Y, respectively on which no separation aioms are assumed unless otherwise stated For a subset A of a space X ClA IJSRST Received : 25 June 208 Accepted : 05 July 208 July-August-208 [ 4 (9) : 8-29] 0

2 and Int A denote the closure and the interior of A respectively Definition 2 Let X, be a topological space and A X Then A is called b open set in X if A Cl Int A Int Cl A The family of all b open subsets of a topological space X, is denoted by BOX, or (Briefly BO X ) Lemma 22 Arbitrary union of b open sets in a topological space is b open set Proof Let X, be a topological space and let A : be a family of b open sets in X Let A A Then for each, A A Then it follows immediately that Cl Int A Cl Int A and Int Cl A Int Cl A Further by A Cl Int A Int Cl A hypothesis, Hence it implies immediately that A Cl Int A Int Cl A, for each A A Cl Int A Int Cl A Thus Therefore, A A is a b open set in X Definition 23 Let X, be a topological space and A X Then A is called Bc open set in X if for each A BOX,, there eists a closed set F such that F X The family of all Bc open subsets of a topological space X, is denoted by or (Briefly Bc OX Bc O X, Bc closed if ) A is called c A X A is Bc open set The family of all Bc closed subsets of a topological space X, is denoted by Bc CX, Bc CX ) or (Briefly Remark 24 It is clear from the definition that every Bc open set is b open, but the converse is not true in general as shown by the following eample Eample 25 Suppose that X, 2, 3, and the, X,, 2,, 2 Then the topology closed sets are: X,, 2, 3,,3, 3 Hence BOX,, X,, 2,, 2,,3, 2,3 and Bc OX,, X,,3, 2,3 Then is b open, but is not Bc open Remark 26 The intersection of two Bc open sets is not Bc open set in general as shown by the following eample Eample 27 Suppose that X, 2, 3, and the topology, X,, 2,, 2 Then,3 and 2,3 are Bc open sets where as, 3 2,3 3 is not Bc open set Theorem 28 Arbitrary union of Bc open sets in a topological space is Bc open set Proof Let A : be a family of Bc open sets in a topological space X, Then for each, A is a b open set in X and by Lemma 22, A is b open set If A, then there eists such that A Since A is Bc open set, there eists a closed set F such that F A A Hence A Bc open set in X Theorem 29 A subset A of a topological space X, is Bc open if and only if A is b open and it is a union of closed sets Proof Let A be a Bc open set Then A is b open and for each A, there eists a closed set F such that F A This implies that A F A Thus A A A F A is where F is closed set for each A The converse is directed from definition of Bc open sets Definition 20 Let X, be a topological space and A X Then Bc closure of A in X is the set International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 9

3 Bc ClA K X : Kis Bc closed and A K Definition 2 Let X, be a topological space and A X Then Bc interior of A in X is the set Bc Int A U X : Uis Bc open and U A Definition 22 Let f : X, Y, be a function Then the subset, f : X X Y is called the graph of f and is denoted by G f III CONTRA Bc CONTINUOUS FUNCTIONS Definition 3 A function :,, called contra Bc continuous if f X Y is f V is Bc closed in X for every open set V of Y Definition 32 Let X, be topological space and A X Then the intersection of all open sets of X containing A is called kernel of A and is denoted by Ker A Lemma 33 The following properties hold for subsets A and B of a topological space X, a Ker A if and only if A F for any closed set F of X containing b A Ker A and A Ker A X c If A B, then if A is open in Ker A Ker B Lemma 34 The following properties hold for a subset A of a topological space X, : i Bc Int A X Bc Cl X A ; ii Bc ClA each Bc OX, ; if and only if A U for iii A is Bc open if and only if A Bc Int A ; iv A is Bc closed if and only if A Bc Cl A Theorem 35 Let :,, f X Y be a function Then the following conditions are equivalent: a f is contra Bc continuous; b for each X and each closed subset F of Y containing, such that f U F ; f there eists UBc OX, c for each closed subset F of Y, Bc open in X; d A of X; f F is f Bc Cl A Ker f A for every subset e Bc B for every subset B of Y Proof a b : Let X and F be any closed set of Y containing f Using (a), we have f Y F X f F is Bc closed so f F is Bc open in X Taking U f F we get U and f U F b c: f F Then in X and Let F be any closed set of Y and f F and there eists a Bc open subset U containing such that f U F Therefore, we obtain : f F U f F which is Bc open in X c a: Y V f Y V X f V Therefore, f V is Bc Let V be any open set of Y Then since is closed in Y, by (c) c d : is Bc open in X closed in X Let A be any subset of X Suppose that y Ker f A Then by Lemma 33, there eists a closed set F of Y containing y such that f A F This implies that A f F and so Bc Cl A f F, Therefore, we International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 20

4 obtain f Bc Cl A F and y f Bc Cl A Hence, f Bc Cl A Ker f A d e : Let B be any subset of Y Using (d) and Lemma 33 we have f Bc Cl f B Ker f f B Ker B Thus it follows that Bc Cl f B f Ker B e a: Let V be an open subset of Y Then from Lemma 33 and (e) we have and Bc Cl f V f Ker V f V hence Bc Cl f V f V f V This shows that is Bc closed in X The following lemma can be verified easily Lemma 36 A function :,, f X Y is Bc continuous if and only if for each X and for each open set V of Y containing f, there eists UBc OX, such that f U V Theorem 37 Suppose that a function f : X, Y, is contra Bc continuous and Y is regular Then f is Bc continuous Proof Let X and V be an open set of Y containing f Since Y is regular, there eists an open set G in Y containing V Cl G f such that Again, since f is contra Bc continuous, so by Theorem 35, there eists U Bc O X, such that f U Cl G Then f U Cl G V Hence f is Bc continuous Definition 38 A function f : X, Y, is called almost Bc continuous if for each X and each open set V of Y containing f, there eists, U Bc O X such that f U Bc Int Cl V Almost Bc continuous function can be equivalently defined as in the following proposition Proposition 39 Let f : X, Y, be a function Then the following statements are equivalent: a f is almost Bc continuous b For each containing, such that f U V c X and each regular open set V of Y f there eists U Bc O X, f V is Bc open in X for every regular open set V of Y Definition 30 A function f : X, Y, is said to be pre Bc open if image of each Bc open set of X is a Bc open set of Y Denition 3 A function f : X, Y, to be is said Bc irresolute if preimage of a Bc open subset of Y is a Bc open subset of X Theorem 32 Suppose that a function f : X, Y, is pre Bc open and contra Bc continuous Then f is almost Bc continuous Proof Let X and V be an open set containing f Since f is contra Bc continuous, then by Theorem 35, there eists UBc OX, such that f U Cl V pre Bc open, Again, since f is f U is Bc open in Y Therefore, f U Bc Int f U and hence f U Bc Int Cl f U Bc Int Cl V So f is almost Bc continuous Theorem 33 Let X, : be any family of topological spaces If a function f : X X is contra Bc continuous, then International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 2

5 f: X X is contra Bc continuous, for each, where onto X is the projection of X Proof For a fied, let V be any open subset of is continuous, V X Since is open in X Since f is contra Bc continuous, f V f V Therefore, is Bc closed in X f is contra Bc continuous for each, Definition 34 Let X, be a topological space Then the Bc frontier of a subset A of X, denoted by Bc Fr A, is defined as Bc Fr A Bc Cl A Bc Cl X A Bc Cl A Bc Int A Theorem 35 The set of all points of X at which f : X, Y, is not contra Bc continuous is identical with the union of Bc frontier of the inverse images of closed sets of Y containing f Proof Necessity: Let f be not contra Bc continuous at a point X Then by Theorem 35, there eists a closed set F of Y containing f such that every U Bc O X,, f U Y F for which implies that U f Y F Therefore, Bc Cl f Y F Bc Cl X f F Again, since f F Bc Cl f F, we get and so it follows that Bc Fr f F Sufficiency: Suppose that Bc Fr f F some closed set F of Y containing for f and f is contra Bc continuous at Then there eists, such that f U F U Bc O X Therefore U f F and hence it follows that Bc Int f F X Bc Fr f F But this is a contradiction So f is not contra Bc continuous at Definition 36 A function f : X, Y, is called almost weakly Bc continuous if, for each X and for each open set V of Y containing f, there eists U Bc O X, such that Cl V f U Theorem 37 Suppose that a function f : X, Y, is contra Bc continuous Then f is almost weakly Bc continuous Proof For any open set V of Y, Cl V is closed in Y f Cl V Since f is contra Bc continuous, is Bc open set in X We take U f Cl V, then f U Cl V weakly Bc continuous Hence f is almost Theorem 38 Let f : X, Y, and g : Y, Z, be any two functions Then the following properties hold: i If f is contra Bc continuous function and g is a continuous function, then gof is contra Bc continuous ii If f is Bc irresolute and g is contra Bc continuous, then gof is contra Bc continuous Proof i For X, let W be any closed set of Z containing g f Since g is continuous, V g W is closed in Y Also, since f is contra Bc continuous, there eists, such that f U V U Bc O X Therefore gf U g f U g V W and so it implies that gf U W Hence, gof is contra Bc continuous International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 22

6 ii For X, let W be any closed set of Z containing g f Since g is contra Bc continuous, there eists V Bc O Y, f W g V such that Again, since f is Bc irresolute, there eists U Bc O X, such that f U V This shows that gf U W Hence, gof is contra Bc continuous Theorem 39 Let f : X, Y, be surjective Bc irresolute and pre Bc open function and :,, g : Y, Z, be any function Then g f X Z is contra Bc continuous if and only if g is contra Bc continuous Proof The if part is easy to prove To prove only if" part, let g f : X, Z, be contra Bc continuous and let F be a closed subset of Z Then g f F ie f g F f f g F pre Bc open, subset of Y and so g F is a Bc open subset of X is Bc open in X Since f is is a Bc open is Bc open in Y Hence, g is contra Bc continuous Definition 320 A topological space X, is said to be Bc normal if each pair of non empty disjoint closed sets can be separated by disjoint Bc open sets Definition 32 A topological space X, is said to be ultranormal if each pair of non empty disjoint closed sets can be separated by disjoint clopen sets Theorem 322 Suppose that f : X, Y, is a contra Bc continuous, closed injection and Y is ultranormal Then X is Bc normal Proof Let A and B be disjoint closed subsets of X Since f is closed injection, f A and f B are disjoint closed subsets of Y Again, since Y is ultranormal, f A and f B are separated by disjoint clopen sets P and Q (say) respectively Therefore, f A P and f B Q A f P and B f Q, where f P f Q ie, and are disjoint Bc open sets of X (since f is contra Bc continuous) This shows that X is Bc normal Definition 323 A topological space X, is called Bc connected provided that X is not the union of two disjoint nonempty Bc open sets of X Theorem 324 Suppose that f : X, Y, contra Bc continuous surjection, where X is Bc connected and Y is any topological space, then Y is not a discrete space Proof If possible, suppose that Y is a discrete space Let P be a proper nonempty open and closed subset Y Then f P of is a proper nonempty Bc open and Bc closed subset of X, which contradicts to the fact that X is Bc connected Hence the theorem follows Theorem 325 Suppose that f : X, Y, is is contra Bc continuous surjection and X is Bc connected Then Y is connected Proof If possible, suppose that Y is not connected Then there eist nonempty disjoint open sets P and Q such that Y P Q So P and Q are clopen sets of Y Since f is contra Bc continuous function, f f P and f Q P and f Q are Bc open sets of X Also are nonempty disjoint Bc open sets of X and X f P f Q, which contradicts to the fact that X is Bc connected Hence Y is connected Theorem 326 A topological space X, is Bc connected if and only if every contra Bc continuous function from X into any space Y, is constant T Proof Let X be Bc connected Now, since Y is a T space, f y : y Y is disjoint International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 23

7 Bc open partition of X If 2 (where denotes the cardinality of ), then X is the union of two nonempty disjoint Bc open sets Since X is Bc connected, we get Hence, f is constant Conversely, suppose that X is not Bc connected and every contra Bc continuous function from X into any T space Y is constant Since X is not Bc connected, there eists a non-empty proper Bc open as well as Bc closed set V (say) in X We consider the space Y 0, with the discrete topology The function f : X, Y, defined by f V 0 and f X V is obviously contra Bc continuous and which is non constant This leads to a contradiction Hence X is Bc connected Definition 327 A topological space X, is said to be Bc T2 if for each pair of distinct points, y in X there eist U Bc O X, V Bc O X, y such that U V Theorem 328 Let and X, and, Y be two topological spaces and suppose that for each pair of distinct points and y in X there eists a function such that f f y f : X, Y, where Y is an Urysohn space and f is contra Bc continuous function at and y Then X is Bc T 2 Proof Let, y X and y Then by assumption, there eists a function f X Y :,,, such that f f y where Y is Urysohn and f is contra Bc continuous at and y Now, since Y is Urysohn, there eist open sets U and V of Y containing f and f y respectively, such that Cl U Cl V Also, f being contra Bc continuous at and y there eist Bc open sets P and Q containing and y respectively such that f P Cl U f Q Cl V Then f P f Q P Q Therefore X is Bc T 2 and and so Corollary 329 If f : X, Y, is contra Bc continuous injection where Y is an Urysohn space, then X is Bc T 2 Corollary 330 If f is contra Bc continuous injection of a topological space X, into an ultra Hausdorff space Y,, then X is Bc T 2 Proof Let, y X where y Then, since f is an injection and Y is ultra Hausdorff, f f y and there eist disjoint closed sets U and V containing f and f y respectively Again, since f is contra Bc continuous, f U Bc O X, and f V Bc O X, y with f U f V Bc T 2 This shows that X is IV ALMOST CONTRA Bc CONTINUOUS FUNCTIONS Definition 4 A function f : X, Y, called almost contra Bc continuous if f V Bc closed for every regular open set V of Y Theorem 42 Let f : X, Y, is is be a function Then the following statements are equivalent: a f is almost contra Bc continuous; b f F is Bc open closed set F of Y; c in X for every regular for each X and each regular open set F of Y containing, such that f U F f there eists U Bc O X, International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 24

8 d for each X and each regular open set V of Y non-containing f, there eists a Bc closed f V K set K of X non-containing such that Proof a b : Then Y F f Y F X f F Bc C X f F Bc O X Let F be any regular closed set of Y is regular open and therefore Hence, The converse part is obvious b c : Let F be any regular closed set of Y containing f Then f F Bc O X and f F Taking U f F we get f U F c b : Let F be any regular closed set of Y and f F Then, there eists U Bc O X, such that f U F and so U f F Also, f F U we have f F f F Bc O X Hence c d : Let V be any regular open set of Y noncontaining f() Then Y V is regular closed set of Y containing f() Hence by (c), there eists U Bc O X, such that f U Y V Hence, we obtain U f Y V X f V and so f V X U Now, since UBc O X, X U is Bc closed set of X not containing The converse part is obvious Theorem 43 Let f : X, Y, be almost contra Bc continuous Then f is almost weakly Bc continuous Proof For X, let H be any open set of Y containing f Then of Y containing f Cl H is a regular closed set Then by Theorem 42, there eists GBc O X, such that f G Cl H So f is almost weakly Bc continuous Theorem 44 Let f : X, Y, be an almost contra Bc continuous injection and Y is weakly Hausdorff Then X is Bc T Proof Since Y is weakly Hausdorff, for distinct points, y of Y, there eist regular closed sets U and V such that f U, f y V, f V contra Bc continuous, f y U and Now, f being almost f U and f V open subsets of X such that f U and y f V, f V Bc y f U shows that X is Bc T Corollary 45 If f : X, Y, are This is a contra Bc continuous injection and Y is weakly Hausdorff, then X is Bc T Theorem 46 Let f : X, Y, be an almost contra Bc continuous surjection and X be Bc connected Then Y is connected Proof If possible, suppose that Y is not connected Then there eist disjoint non-empty open sets U and V of Y such that Y U V Since U and V are clopen sets in Y, they are regular open sets of Y Again, since f is almost contra Bc continuous surjection, f U and f V X and X f U f V are Bc open sets of This shows that X is not Bc connected But this is a contradiction Hence Y is connected Definition 47 A topological space X, is said to be Bc compact if every Bc open cover of X has a finite subcover Definition 48 A topological space X, is said to be countably Bc compact if every countable cover of X by Bc open sets has a finite subcover Definition 49 A topological space X, is said to be Bc Lindeloff if every Bc open cover of X has a countable subcover, International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 25

9 Theorem 40 Let f : X, Y, be an almost contra Bc continuous surjection Then the following statements hold: a If X is Bc compact, then Y is S closed b If X is Bc Lindeloff, then Y is S Lindeloff c If X is countably Bc compact, then Y is countably S closed Proof a : Let V : I be any regular closed cover of Y Since f is almost contra Bc continuous, then f V : I is a Bc open cover of X Again, since X is Bc compact, there eist a finite subset I 0 of I such that X f V : I and hence Y V : I 0 The proofs of b Therefore, Y is S closed a : omitted and c 0 are being similar to Definition 4 A topological space X, is said to be Bc closed compact if every Bc closed cover of X has a finite subcover Definition 42 A topological space X, is said to be countably Bc closed if every countable cover of X by Bc closed sets has a finite subcover Definition 43 A topological space X, is said to be Bc closed Lindeloff if every Bc closed cover of X has a countable subcover Theorem 44 Let f : X, Y, be an almost contra Bc continuous surjection Then the following statements hold: a If X is Bc closed compact, then Y is nearly compact b If X is Bc closed Lindeloff, then Y is nearly Lindeloff c If X is countably Bc closed compact, then Y is nearly countable compact Proof a :Let V : I be any regular open cover of Y Since f is almost contra Bc continuous, then f V : I is a Bc closed cover of X Again, since X is Bc closed compact, there eists a finite subset I 0 of I such that X f V : I0 and hence 0 Y V : I Therefore, Y is nearly compact The proofs of b a : omitted and c are being similar to V CLOSED GRAPHS VIA Bc OPEN SETS Definition 5 Let f : X, Y, Then the graph, : be a function G f f X of f is said to be Bc closed (resp contra Bc closed ) if for each y X Y G f,, there eist a U Bc O X, and an open set (resp a closed set) V in Y containing y such that U V G f Lemma 52 A graph G f of a function f : X, Y, is Bc closed (resp contra Bc closed) in X Y if and only if for each y X Y G f U Bc O X,,, there eist and an open set (resp a closed set) V in Y containing y such that f U V Proof We shall prove that f U U V G f Let U V G f V Then there eists, yu V and y G f This implies that U, y V, and y f V Therefore, f U V Hence the result follows Theorem 53 Suppose that f : X, Y, is contra Bc continuous and Y is Urysohn Then G f is contra Bc closed in X Y International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 26

10 Proof Let y X Y G f y f, It follows that Since Y is Urysohn, there eist open sets V and W in Y such that f V, y W and Cl V Cl W Now, since f is contra Bc continuous, there eists a U Bc O X, such that f U Cl V which implies that f U Cl W Hence by Lemma 52, G f is contra Bc closed in X Y Theorem 54 Let f : X, Y, be a function and g : X X Y be the graph function of f, defined by g, f for every X If g is contra Bc continuous, then f is contra Bc continuous Proof Let G be an open set in Y, then X G is an open set in X Y Since g is contra Bc continuous, it implies that f G g X G is a Bc closed set of X Therefore, f is contra Bc continuous Theorem 55 Let f : X, Y, have a contra Bc closed graph If f is injective, then X is Bc T Proof Let and 2 be any two distinct points of X Then, we have f 2 X Y G f, Then, there eists a Bc open set U in X containing and F C Y, f such that f U F Hence 2 U f F Therefore, we have 2 U This implies that X is Bc T Definition 56 The graph G f of a function f : X, Y, is said to be strongly contra Bc closed if for each y X Y G f there eist U Bc O X,,, and regular closed set V in Y containing y such that U V G f Lemma 57 The graph G f of a function f : X, Y, is strongly contra Bc closed in X Y if and only if for each y X Y G f U Bc O X,,, there eist and regular closed set V in Y containing y such that f U V Theorem 58 Let f : X, Y, be an almost weakly Bc continuous and Y is Urysohn Then G f is strongly contra Bc closed in X Y Proof Let, y X Y G f Then y f and since Y is Urysohn, there eist open sets G, H in Y such that f G, y H and Cl G Cl H Now, since f is almost weakly Bc continuous, there eists U Bc O X, such that f U Cl G This implies that f U Cl H, where Cl Int H f U Cl Int H is regular closed in Y Hence by above Lemma 57, G f is strongly contra Bc closed in X Y Theorem 59 Let f : X, Y, Bc continuous and Y is 2 contra Bc closed in X Y T Then be an almost G f is strongly Proof Let, y X Y G f Then y f and since Y is T, 2 there eist open sets G and H containing y and f, respectively, such that G H ; which is equivalent to Cl G Int Cl H Again, since f is almost Bc continuous and Int Cl H is regular open, by proposition 39, there eists W Bc O X, such that f W Int Cl H This implies that f W Cl G and by Lemma 57, G f strongly contra Bc closed in X Y is International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 27

11 Definition 50 A filter base on a topological space X, is said to be Bc convergent to a point in X if for any U Bc O X, containing, there eists an F such that F U Theorem 5 Prove that every function : X, Y,, where, Y is compact with Bc closed graph is Bc continuous Proof Let be not Bc continuous at X Then T S for every T Bc O X, It there eists an open set S in Y containing such that is obvious to verify that T X : T Bc O X, is a filterbase on X that Bc converges to Now we shall show that T Y S: T Bc O X, is a filterbase on Y Here for every T Bc O X T S, ie T Y S,, So Let G, H Then there are T, T2 such that and H T Y S G T Y S 2 Since is a filterbase, there eists a T3 such that and so T3 T T2 W T3 Y S with W G H It is clear that G and G H imply H Hence is a filterbase on Y Since Y S is closed in compact space Y, S is itself compact So, y Y S must adheres at some point Here y y G ensures that, Thus Lemma 52 gives us a U Bc O X, and an open set V in Y containing y such that U V, i e U Y S V But this is a contradiction Theorem 52 Suppose that an open surjection X Y :,, possesses a Bc closed graph Then Y is T 2 Proof Let p, p2 Y with p p 2 Since is a surjection, there eists an p and, p2g U Bc O X, p such that U V 2 p 2 X such that Therefore and so by Lemma 52, there eist and open set V in Y containing U Since is Bc open, and V are disjoint open sets containing p and p 2 respectively So Y is T 2 Corollary 53 If a function : X, Y, is a surjection and possesses a Bc closed graph, then Y is T VI ACKNOWLEDGMENT The first author is highly and greatly thankful to Prince Mohammad Bin Fahd University for providing all necessary research facilities during the preparation of this research paper VII REFERENCES [] Raad Aziz Al-Abdulla and Ruaa Muslim Abed, On Bc open sets, International Journal of Science and Research, Volume 3, Issue 0, (204), 7 pages [2] Metin Akdag & Alkan Ozkan, Some Properties of Contra gb continuous Functions, Journal of New Results in Science, (202) [3] D Andrijevic, On b open sets, Math Vesnik, 48(996), [4] G Anitha and M Mariasingam, Contra g ~ -WG Continuous Functions, International Journal of Computer Applications ( ) Volume 49 No, (July 202), [5] S Balasubramanian, Almost contra vg-continuity, International Journal of Mathematical Engineering and Science, Volume Issue 8 (August 202), 5 65 [6] M Caldas and S Jafari, Some properties of contra continuous functions, Mem Fac Sci Kochi Univ, Series A Math, 22(200), 9 28 International Journal of Scientific Research in Science and Technology (wwwijsrstcom) 28

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