UPPER AND LOWER WEAKLY LAMBDA CONTINUOUS MULTIFUNCTIONS

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1 UPPER AND LOWER WEAKLY LAMBDA CONTINUOUS MULTIUNCTIONS R.VENNILA Department of Mathematics, SriGuru Institute of Technology,Varathaiyangar Palayam, Coimbatore , India Mathematics Subject Classification: 54C10, 54D10. ABSTRACT: In this paper, a new notion in topological spaces called weakly -continuous multifunction is introduced and studied the characterization of weakly -continuous multifunctions. Keywords: Topological spaces, -open sets, -closed sets, weakly -continuous multifunctions. 1. INTRODUCTION AND PRELIMINARIES It is well known that various types of functions play a significant role in the theory of classical point set topology. A great number of papers dealing with such functions have appeared, and good numbers of them have been extended to multifunctions. This implies that both functions and multifunctions are important tools in the whole Mathematical Science. Maki [6] introduced the notion of -sets in topological spaces. A -set is a set A which is equal to its kernel, that is, to the intersection of all open super sets of A. Arenas et.al. [1] introduced and investigated the notion of -closed sets by involving -sets and closed sets. Caldas et.al. [3] introduced and studied some new notions by utilizing the notion of -open sets defined in [1]. Let A be a subset of a topological space (X,). The closure and the interior of a set A is denoted by Cl(A), Int(A) respectively. A subset A of a topological space (X,) is said to be -closed [1] if A = BC, where B is a -set and C is a closed set of X. The complement of -closed set is called -open [3]. A point xx in a topological space (X,) is said to be -cluster point of A [3] if for every - R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 267

2 open set U of X containing x, A U. The set of all -cluster points of A is called the -closure of A and is denoted by Cl (A) [3]. A point xx is said to be the -interior point of A if there exists a -open set U of X containing x such that U A. The set of all -interior points of A is said to be the -interior of A and is denoted by Int (A). The family of all -open (resp.-closed) sets of X is denoted by O(X) (resp. C(X)). The family of all -open (resp. -closed) sets of a space (X,) containing the point xx is denoted by O(X, x) ( resp. C(X, x)). Let B be a subset of Y. or a multifunction Y, upper and lower inverse of any subset B of Y is defined by B X: x B and B xx: x B. 2. WEAKLY -CONTINUOUS MULTIUNCTIONS X x Definition 2.1: A multifunction X Y is said to be: (i) upper weakly -continuous if for each x V, there exists U OX, x such that U ClV (ii) lower weakly -continuous if for each x X and each open set V of Y such that ; x V, there exists U OX, x such that U ClV x X and each open set V of Y such that ; (iii) Weakly -continuous if is both upper weakly -continuous and lower weakly -continuous. Theorem 2.2: or a multifunction X Y, the following statements are equivalent: (i) is upper weakly -continuous; IntK K (ii) V Int ClV (iii) Cl (iv) for each for any open set V of Y; for any closed set K of Y; x Xand each open set V containing x, there exists a -open set U containing x such that U V Cl. Proof: (i)(ii): Let V be any open set and containing x such that U ClV. Hence, Int ClV x V. By (i), there exists a -open set U x. R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 268

3 (ii)(i): Let V be any open set and ClV. Take U Int ClV Int continuous. x V. By (ii), x V ClV. Thus, we obtain that is upper weakly - (ii)(iii): Let K be any closed set of Y. Then, YKis an open set By (ii), Y K X K Int ClY K Cl Int K K (i)(iv): Obvious.. The converse is similar. Int Y IntK. Thus, Definition 2.3: [4] A multifunction X Y is said to be: (i) upper -continuous at a point there exists U OX, x such that U V (ii) lower -continuous at a point there exists OX, x x X if for each open set V of Y such that ( x ) V, ; x X if for each open set V of Y such that ( x ) V, U such that (u) V for every u U ; (iii) upper (lower) -continuous if has this property at each point of X. Remark 2.4: It is clear that every upper -continuous multifunction is upper weakly -continuous. But the converse is not true in general, as the following example shows. Example 2.5: Let X a, b, c, τ, a, b, (X,τ) (X,τ) but not upper -continuous. defined by a a, b c, c b X. Then the multifunction is upper weakly -continuous Theorem 2.6: or a multifunction X Y, the following statements are equivalent: (i) is lower weakly -continuous; IntK K (ii) V Int ClV (iii) Cl (iv) for each for any open set V of Y; for any closed set K; x X and each open set V such that x V, there exists a -open set U for each u U. containing x such that u ClV R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 269

4 Proof: It is similar to the proof of the Theorem 2.2. Theorem 2.7: Let X Y be a multifunction such that x is an open set of Y for each x X. Then is lower -continuous if and only if lower weakly -continuous. Proof: Let x Xand V be an open set of Y such that x V ). Then there exists a -open for each u U set U containing x such that u ClV for each. Since u is open, u V u U and hence is lower -continuous. The converse follows by Remark 2.4. Definition 2.8: A topological space X, τ is said to be normal if for every disjoint closed sets V 1 and V2 of X, there exist two disjoint open sets U1and U 2 such that V1 U1 and V2 U2. Theorem 2.9: If V X Y is upper weakly -continuous and satisfies the condition ClV for every open set V of Y, then is upper -continuous. Proof: Let V be any open set of Y. Since is weakly -continuous, we have V Int ClV and hence V Int -open and it follows that is upper -continuous. Cl V V. Thus, V Int is Theorem 2.10: Let X Y be a multifunction such that x is closed in Y for each x Xand Y is normal. Then is upper weakly -continuous if and only if is upper -continuous. Proof: Suppose that is upper weakly -continuous. Let x. Since x x X and G be an open set containing is closed in Y and Y is normal, there exist open sets V and W such that x V, X G W and W V. We have x V ClV ClX W X W G. Since is upper weakly -continuous, there exists a -open set U containing x such that V Cl G. This shows that is upper -continuous. The converse follows by Remark 2.4. U Lemma 2.11 : [5] If A is an α -paracompact α -regular set of a topological space X, τ and U an open neighbourhood of A, then there exists an open set G of X such that A G ClG U. R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 270

5 or a multifunction X Y, by Cl X Y we denote a multifunction as follows: x x Cl Cl for each x X. Similarly, we denote Cl. Lemma 2.12: If each X Y is a multifunction such that x is α -paracompact α -regular for x X, then we have the following: (i) G V V (ii) G V V for each open set V of Y, for each closed set V of Y, where G denotes Cl or Cl. Proof: (i): Let V be any set of Y and have x G V. Then Gx V and x Gx V x V and hence G V V. Then we have x V. We and by Lemma 2.11 there exists an open set V such that x H ClH V. Since V G V, G V V (ii): ollows from (i). Lemma 2.13: or a multifunction X Y, we have the following: (i) G V V (ii) G V V Theorem 2.14: Let for each open set V of Y, for each open set V of Y, where G denotes Cl or Cl.. X Y be a multifunction such that x is α -regular and α -paracompact for every x X. Then the following properties are equivalent: (i) is upper weakly -continuous; (ii) Cl is upper weakly -continuous; (iii) (iv) scl is upper weakly -continuous; Cl is upper weakly -continuous; (v) αcl is upper weakly -continuous; (vi) pcl is upper weakly -continuous. Proof: We put G = Cl, scl, Cl, αclor pcl in the sequel. Necessity: Suppose that is upper weakly -continuous. Then it follows by Theorem 2.2 and Lemmas 2.12 and 2.13 that for every open set V of Y containing x, G V By Theorem 2.2, G is upper weakly -continuous. V ClV G ClV R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 271 Int Int.

6 Sufficiency: Suppose that G is upper weakly -continuous. Then it follows by Theorem 2.2 and Lemmas 2.12 and 2.13 that for every open set V of Y containing Int G Cl -continuous. V Int Cl V G x, V G V. It follows by Theorem 2.2 that is upper weakly Definition 2.15: Let A be a subset of a topological space X. The -frontier [3] of A, denoted by r A, is defined by r A Cl A Cl X A Theorem 2.16: Let. X Y be a multifunction. The set of all points x of X such that is not upper weakly -continuous (resp. lower weakly -continuous) is identical with the union of -frontiers of the upper (resp. lower) inverse images of the closure of open sets containing (resp. meetings) x. Proof: Let x be a point of X at which is not upper weakly -continuous. Then there exists an open set V containing x such that X ClV U for every -open set U containing x. Therefore, x X ClV. Since x V, we have Cl ClV x r Cl V containing U V Cl x and hence. Conversely, if is upper weakly -continuous at x, then for every open set x, there exists a -open set U containing x such that U ClV ClV. Therefore, we obtain U Int ClV V x Int Cl. hence x. This contradicts that Theorem 2.17: Let and G be respectively upper weakly -continuous and upper weakly continuous multifunctions from a topological space X, τto a strongly normal space Y, σ. Then the set x : x Gx Proof: Let K is -closed in X. x X K, then x Gx. Since and G are point closed multifunctions and Y is a strongly normal space, there exist disjoint open sets U and V containing respectively we have Cl U ClV x and x G,. Since and G are upper weakly -continuous functions, then there exist -open set U 1 containing x and open set U 2 containing x such that R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 272

7 U 1 ClV and U ClV x and H K ; hence K is -closed in X. Definition 2.18: or a multifunction 2. Now set H U 1 U2, then H is an -open set containing is defined as follows: x x x X Y, the graph multifunction : X X Y for every x X and the subset x x: x X X Y is called the graph multifunction of and is denoted by x Lemma 2.19: or a multifunction X Y, the following holds: (i) A B A B ; (ii) A B A B Theorem 2.20: Let for any subset A of X and B of Y. G. X Y be a multifunction and X be a connected space. If the graph multifunction of is upper (lower) weakly -continuous, then is upper (lower) weakly -continuous. Proof: Let relative to x X and V be any open subset of Y containing x XYand X V. Since XV is an open set x, there exists a -open set U containing x such that U ClX V XClV. By Lemma 2.19, we have U G XClV ClV and U V Cl. Thus, is upper weakly -continuous. The proof of the lower weakly -continuity of can be done by the same token. REERENCES [1].G.Arenas, J.Dontchev and M.Ganster, On -closed sets and dual of generalized continuity, Q&A Gen.Topology, 15, 3-13, [2] C. Berg, Espaces topologiques functions multivoques, paris, Dunod [3] M.Caldas, S.Jafari and G.Navalagi, More on -closed sets in topological spaces, Revista Columbiana de Matematicas, 41(2), , [4] M. Caldas, E. Hatir, S. Jafari and T.Noiri, A new Kupka Type Continuity, -Compactness and Multifunctions CUBO A Mathematical Journal 11, 2, 1-13, R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 273

8 [5] I. Kovacevic, Subsets and paracompactness, Univ.u. Novom Sadu Zb. Rad. Prirod. Mat. ak. Ser. Mat., 14, 79-87, [6] H.Maki, 1.Oct, Generalized -sets and the associated closure operator, The special issue in commemoration of Prof. Kazusada IKEDA s Retirement, , R S. Publication (rspublication.com), rspublicationhouse@gmail.com Page 274