scl(a) G whenever A G and G is open in X. I. INTRODUCTION

Transcription

1 Generalized Preclosed Sets In Topological Spaces 1 CAruna, 2 RSelvi 1 Sri Parasakthi College for Women, Courtallam Matha College of Arts & Science, Manamadurai rajlakh@gmailcom Abstract In th paper, we introduce a new class of sets called -generalized preclosed sets and -generalized preopen sets in topological spaces and study some of their properties Index terms- I INTRODUCTION In 1970, Levine[6] introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces Using generalized closed sets, Dunham[5] introduced the concept of the closure operator c1* and a new topology and studied some of their properties SPArya[2], PBhattacharyya and BKLahiri[3], JDontchev[4], HMaki, RDevi and KBalachandran[9], [10], PSundaram and APushpalatha[12], ASMashhour, MEAbd E1-Monsof and SNE1-Deeb[11], DAndrijevic[1] and SMMaheshwari and PCJain[9], Ivan Reilly [13], APushpalatha, SEswaran and PRajaRubi [14] introduced and investigated generalized semi closed sets, semi generalized closed sets, generalized semi preclosed sets, generalized closed sets, generalized- closed sets, strongly generalized closed sets, preclosed sets, semi-preclosed sets and closed sets, generalized preclosed sets and -generalized closed sets respectively In th paper, we obtain a new generalization of preclosed sets in the weaker topological space(x, ) Throughout th paper X and Y topological spaces on which no separation axioms assumed unless otherwe explicitly stated For a subset A of a topological space X, int(a),, (A),, and denote the interior, closure,, semi-closure, semipreclosure, -closure, preclosure and complement of A respectively 2Preliminaries: We recall the following definitions Definition: 21 A subset A of a topological space (X, ) called (i) Generalized closed (briefly g-closed)[6] if cl(a) G whenever A G and G open in X (ii) Semi-generalized closed(briefly sg-closed)[3] if scl(a) G whenever A G and G semi open in X (iii) Generalized semi-closed (briefly gs-closed)[2] if scl(a) G whenever A G and G open in X (iv) α- closed[8] if cl(int(cl(a))) A (v) α-generalized closed (briefly αg-closed)[9] if (A) G whenever A G and G open in X (vi) Generalized α-closed (briefly gα-closed)[10] if spcl(a) G whenever A G and G open in X (vii) Generalized semi-preclosed (briefly gsp-closed)[2] if scl(a) G whenever A G and G open in X (viii) Strongly generalized closed (briefly strongly g- closed)[12] if whenever (ix) Preclosed[11] if (x) Semi-closed[7] if and G g-open in X (xi) Semi-preclosed (briefly sp-closed)[1] if (xii) Generalized preclosed (briefly gp-closed)[13] if whenever and G open The complements of the above mentioned sets called their respective open sets Definition: 22 For the subset A of a topological X, the generalized closure operator defined by the intersection of all g-closed sets containing A Definition: 23 For the subset A of a topological X, the topology by defined Definition: 24 For the subset A of a topological X, (i) the semi-closure of A defined as the intersection of all semi-closed sets containing A 99 P a g e

2 (ii) the semi-pre closure of A defined as the intersection of all semi-preclosed sets containing A (iii) the of A defined as the intersection of all sets containing A (iv) the preclosure of A, denoted by smallest preclosed set containing A Definition: 25 A subset A of a topological space X called closed set ( [14] if the generalized whenever and G The complement of closed set called the open set 3 Preclosed Sets In Topological Spaces In th section, we introduce the concept of - generalized preclosed sets in topological spaces Definition: 31 A subset A of a topological space X called preclosed ( if whenever and G open The complement of preclosed set called the preopen set (briefly Example: 32 Let and let Here (X, ) -generalized preclosed Theorem: 33 Every closed set in X Let A be a closed set Let Since A closed, Therefore A But Thus, we have whenever and G Theorem: 34 Every set in X Let A be a set Let where G Since A, Thus, we have whenever and G Theorem: 35 Therefore A closed Every g-closed set in X a conversely Let A be a g-closed set Assume that gp-closed set but not, G in X Then Since A g-closed But Hence A Therefore The converse of the above theorem need not be true as seen from the following example Example: 36 Consider the topological space with topology -closed but not -closed Remark: 37 The following example shows that Then, the set {a,c} sets independent from sp-closed, sg-closed set,, preclosed set, gs-closed set, gsp-closed set, set and g set Example: 38 Let and be the topological spaces (i) Consider the topology but not spclosed (ii) Consider the topology and sp-closed but not (iii) Consider the topology but not sg-closed (iv) Consider the topology Then the sets sg-closed but not (v) Consider the topology not (vi) Consider the topology set but not (vii) Consider the topology preclosed but Then the but not 100 P a g e

3 (viii) Consider the topology set pre-closed but not (ix) Consider the topology (x) Consider the topology closed (xi) Consider the topology but not gs-closed Then the but not Then the sets but not closed (xii) Consider the topology (xiii) Consider the topology (xiv)consider the topology not (xv) Consider the topology but not gsp- closed but not g but not g-closed but (xvi)consider the topology Theorem: 39 For any two sets A and B Since A Therefore the closed sets Therefore, we have but not and Since we have Also, and also a closed set Again, A and B, Implies Thus, containing Thus, a closed set containing Since the smallest closed set Theorem: 310 Union of two We have sets in X a Let A and B be two Let where G Since A and B But by theorem 312 Ther efore Hence a set Theorem: 311 A subset A of X X Let A be a set in set Suppose that F a nonempty subset of Now, F Then F Since Therefore F and F Since a -open set and A a = That That, a contradiction Thus, contain no nonempty set in X Conversely, assume that set Let, G Suppose that 101 P a g e

4 not contained in G then contradiction Therefore, A Corollary: 312 A subset A of X a non-empty set of which a and lence if and only closed set in X The proof follows from the theorem 311 and the fact that every closed set set in X Corollary: 313 A subset A of X open set in X The proof follows from the theorem 311 and the fact that every open set set in X Theorem: 314 If a subset A of X and Let A be a in Then A Suppose A g-closed in X Then, and so which in X Conversely, suppose in X Since A, by the theorem 311, in X Then, Hence, A g-closed Remark 317 From the above dcussion, we obtain the following implications Closed g-closed set then B set in X Let A be a set such that Let be a set of X such that B Since A we have Now, That, U Therefore B set in X The converse of the above theorem need not be true as seen from the following example Example: 315 Consider the topological space with topology Let Then A and B sets in But Theorem: 316 not a subset of REFERENCES [1] DAndrijevic, Semi-preopen sets, MatVesnik, 38 (1986), [2] SPArya and TNour, Characterizations of s-normal spaces, Indian JPure ApplMath, 21(1990), [3] PBhattacharyya and BKLahiri, Semi generalized closed sets in topology, Indina JMath, 29(1987), [4] JDontchev, On generalizing semipreopen sets,mem, Fac SciKochi UniSer A,Math,16(1995), [5] WDunham, A new closure operator for non-t1 topologies, Kyungpook MathJ22(1982),55-60 [6] NLevine, Generalized closed sets in topology, RendCircMatPalermo, 19,(2)(1970),89-86 [7] NLevine, Semi-open sets and semi-continuity in topological spaces AmerMathMonthly; 70(1963), P a g e

5 [8] SNMaheswari and PCJain, Some new mappings, Mathematica, Vol24(47) (1-2)(1982), [9] HMaki, RDevi and KBalachandran, Assicated topologies of generalized α-closed sets and α- generalized closed sets, Mem Fac Sci Kochi Univ(Math)15(1994),51-63 [10] HMaki, RDevi and KBalachandran, Generalized α- closed sets in topology, Bull Fukuoka Uni, EdPart III,42(1993), [11] ASMashhour, MEAbd E1-Monesf and SNE1- Deeb, On precontinuous and weak precontinuous functions, ProcMathPhys SocEgypt 53 (1982),47-53 [12] PSundaram, APushpalatha, Strongly generalized closed sets in topological spaces, Far East JMath Sci(FJMS) 3(4)(2001), [13] Ivan Reilly, Generalized Closed Sets: A survey of recent work, Auckland Univ 1248, 2002(1-11) [14] APushpalatha, SEswaran and PRajaRubi, - generalized closed sets in topological spaces, WCE 2009, July 1-3, 2009, London, UK 103 P a g e