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ISSN: 2277-9655 Impact Factor: 4.116 [Selvi* et al., 5(8): August, 2016] ICโข Value: 3.00 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY ฯ*- GENERALIZED SEMICLOSED SETS IN TOPOLOGICAL SPACES R. Selvi*, C. Aruna * Sri Parasakthi College for Women, Courtallam โ 627802 Research Scholar, Research centre of mathematics, Sri Parasakthi College for Women, Courtallam627802 DOI: 10.5281/zenodo.60107 ABSTRACT In this paper, we introduce a new class of sets called ๐ โ -generalized semiclosed sets and ๐ โ -generalized semiopen sets in topological spaces and study some of their properties. KEYWORDS: ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ set, ๐ โ โ ๐๐ ๐๐๐๐ set 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 ๐ โ and studied some of their properties. S.P.Arya[2], P.Bhattacharyya and B.K.Lahiri[3], J.Dontchev[4], H.Maki, R.Devi and K.Balachandran[9], [10], P.Sundaram and A.Pushpalatha[12], A.S.Mashhour, M.E.Abd E1Monsof and S.N.E1-Deeb[11], D.Andrijevic[1] and S.M.Maheshwari and P.C.Jain[9], Ivan Reilly [13], A.Pushpalatha, S.Eswaran and P.RajaRubi [14] introduced and investigated generalized semi closed sets, semi generalized closed sets, generalized semi preclosed sets, ๐ผ โgeneralized closed sets, generalized- ๐ผ closed sets, strongly generalized closed sets, preclosed sets, semi-preclosed sets and ๐ผ โclosed sets, generalized preclosed sets and ๐ โ -generalized closed sets respectively. In this paper, we obtain a new generalization of preclosed sets in the weaker topological space(X, ๐ โ ). Throughout this paper X and Y are topological spaces on which no separation axioms are assumed unless otherwise explicitly stated. For a subset A of a topological space X, int(A), ๐๐(๐ด), ๐๐ โ (A), ๐ ๐๐(๐ด), ๐ ๐๐๐(๐ด), ๐๐๐ผ (๐ด), ๐๐๐ (๐ด) and ๐ด๐ denote the interior, closure, ๐๐๐๐ ๐ข๐๐ โ , semi-closure, semi-preclosure, ๐ผ-closure, preclosure and complement of A respectively. PRELIMINARIES We recall the following definitions Definition: 2.1 A subset A of a topological space (X , ๐) is called (i) Generalized closed (briefly g-closed)[6] if cl(A)โG whenever AโG and G is open in X (ii) Semi-generalized closed(briefly sg-closed)[3] if scl(A)โG whenever AโG and G is semi open in X. (iii) Generalized semi-closed (briefly gs-closed)[2] if scl(A)โG whenever AโG and G is open in X. (iv) ฮฑ- closed[8] if cl(int(cl(A))) โ A (v) ฮฑ-generalized closed (briefly ฮฑg-closed)[9] if ๐๐๐ผ (A) โ G whenever AโG and G is open in X. (vi) Generalized ฮฑ-closed (briefly gฮฑ-closed)[10] if spcl(A) โ G whenever AโG and G is open in X. http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology [621] ISSN: 2277-9655 Impact Factor: 4.116 [Selvi* et al., 5(8): August, 2016] ICโข Value: 3.00 (vii) Generalized semi-preclosed (briefly gsp-closed)[2] if scl(A) โ G whenever A โ G and G is open in X. (viii) Strongly generalized closed (briefly strongly g-closed)[12] if ๐๐(๐ด) โ ๐บ whenever ๐ด โ ๐บ and G is g-open in X. (ix) Preclosed[11] if ๐๐(๐๐๐ก(๐ด)) โ ๐ด (x) Semi-closed[7] if ๐๐๐ก(๐๐(๐ด)) โ ๐ด (xi) Semi-preclosed (briefly sp-closed)[1] if ๐๐๐ก(๐๐(๐๐๐ก(๐ด))) โ ๐ด. (xii) Generalized preclosed (briefly gp-closed)[13] if ๐๐๐ ๐ด โ ๐บ whenever ๐ด โ ๐บ and G is open. The complements of the above mentioned sets are called their respective open sets. Definition: 2.2 For the subset A of a topological X, the generalized closure operator ๐๐ โ [5] is defined by the intersection of all g-closed sets containing A. Definition: 2.3 For the subset A of a topological X, the topology ๐ โ is defined by ๐ โ = {๐บ: ๐๐ โ (๐บ ๐ ) = ๐บ ๐ }. Definition: 2.4 For the subset A of a topological X, (i) the semi-closure of A(๐๐๐๐๐๐๐ฆ ๐ ๐๐(๐ด))[7] is defined as the intersection of all semi-closed sets containing A. (ii) the semi-Pre closure of A(๐๐๐๐๐๐๐ฆ ๐ ๐๐๐(๐ด))[1] is defined as the intersection of all semipreclosed sets containing A. (iii) the ๐ผ โ ๐๐๐๐ ๐ข๐๐ of A (๐๐๐๐๐๐๐ฆ ๐๐๐ผ (๐ด))[8] is defined as the intersection of all ๐ผ โ ๐๐๐๐ ๐๐ sets containing A. (iv) the preclosure of A, denoted by ๐๐๐ (๐ด)[13], is the smallest preclosed set containing A. Definition: 2.5 A subset A of a topological space X is called ๐ โ generalized closed set (๐๐๐๐๐๐๐ฆ ๐ โ โ ๐๐๐๐๐ ๐๐)[14] if ๐๐ โ (๐ด) โ ๐บ whenever ๐ด โ ๐บ and G is ๐ โ โ ๐๐๐๐. The complement of ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ closed set is called the ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ open set (๐๐๐๐๐๐๐ฆ ๐ โ โ ๐๐๐๐๐). Definition: 2.6 A subset A of a topological space X is called ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ preclosed (๐๐๐๐๐๐๐ฆ ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐)[15] if ๐๐ โ (๐๐๐ (๐ด)) โ ๐บ whenever ๐ด โ ๐บ and G is ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ open. The complement of ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ preclosed set is called the ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ preopen set (briefly ๐ โ โ ๐๐ โ ๐๐๐๐). ๐โ โ ๐ฎ๐๐๐๐๐๐๐๐๐๐ SEMICLOSED SETS IN TOPOLOGICAL SPACES In this section, we introduce the concept of ๐ โ - generalized semiclosed sets in topological spaces. Definition: 3.1 A subset A of a topological space X is called ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ semiclosed (๐๐๐๐๐๐๐ฆ ๐ โ โ ๐๐ ๐๐๐๐ ๐๐) if ๐๐ โ (๐ ๐๐(๐ด)) โ ๐บ whenever ๐ด โ ๐บ and G is ๐ โ โ ๐pen. The complement of ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ semiclosed set is called the ๐ โ โ ๐๐๐๐๐๐๐๐๐ง๐๐ semiopen set (briefly ๐ โ โ ๐๐ ๐๐๐๐). Example: 3.2 Let ๐ = {๐, ๐, ๐} and let ๐ = {๐, ๐, {๐}, {๐, ๐}, {๐}, {๐, ๐}}. Here (X, ๐ โ ) is ๐ โ -generalized semiclosed Theorem: 3.3 Every closed set in X is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐. Proof: Let A be a closed set. Let ๐ด โ ๐บ. Since A is closed, ๐๐(๐ด) = ๐ด โ ๐บ. But ๐๐ โ (๐ ๐๐(๐ด)) โ ๐๐(๐ด). Thus, we have ๐๐ โ (๐ ๐๐(๐ด)) โ ๐บ whenever ๐ด โ ๐บ and G is ๐ โ โ ๐๐๐๐. Therefore A is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐. Theorem: 3.4 Every ๐ โ โ ๐๐๐๐ ๐๐ set in X is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐. Proof: http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology [622] ISSN: 2277-9655 Impact Factor: 4.116 [Selvi* et al., 5(8): August, 2016] ICโข Value: 3.00 Let A be a ๐ โ โ ๐๐๐๐ ๐๐ set. Let ๐ด โ ๐บ where G is ๐ โ โ ๐๐๐๐ Since A is ๐ โ โ ๐๐๐๐ ๐๐, ๐๐ โ (๐ ๐๐(๐ด)) = ๐ด โ ๐บ. Thus, we have ๐๐ โ (๐ ๐๐(๐ด)) โ ๐บ whenever ๐ด โ ๐บ and G is ๐ โ โ ๐๐๐๐. Therefore A is ๐ โ โ ๐๐ closed. Theorem: 3.5 Every g-closed set in X is a ๐ โ โgsclosed set but not conversely. Proof: Let A be a g-closed set. Assume that ๐ด โ ๐บ, G is ๐ โ โ ๐๐๐๐ in X. Then ๐๐(๐ด) โ ๐บ, Since A is g-closed. But ๐๐ โ (๐ ๐๐(๐ด)) โ ๐๐(๐ด). Therefore ๐๐ โ (๐ ๐๐(๐ด)) โ ๐บ. Hence A is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐. The converse of the above theorem need not be true as seen from the following example. Example: 3.6 Consider the topological space ๐ = {๐, ๐, ๐} with topology ๐ = {๐, ๐, {๐}, {๐, ๐}, {๐}, {๐, ๐}}. Then, the set {a,c} is ๐ โ โ ๐๐-closed but not ๐-closed. Remark: 3.7 The following example shows that ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐ sets are independent from sp-closed, sg-closed set, ๐ผ โ ๐๐๐๐ ๐๐ ๐ ๐๐ก, preclosed set, gs-closed set, gsp-closed set, ๐ผ๐ โ ๐๐๐๐ ๐๐ set and g๐ผ โ ๐๐๐๐ ๐๐ set. Example: 3.8 Let ๐ = {๐, ๐, ๐} and ๐ = {๐, ๐, ๐, ๐} be the topological spaces. (i) Consider the topology ๐ = {๐, ๐, {๐}}. Then the sets {๐}, {๐, ๐} ๐๐๐ {๐, ๐} are ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐ but not sp-closed. (ii) Consider the topology ๐ = {๐, ๐, {๐, ๐}}. Then the sets {๐} and {๐} are sp-closed but not ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐. (iii) Consider the topology ๐ = {๐, ๐}. Then the sets {๐}, {๐}, {๐}, {๐, ๐}, {๐, ๐}๐๐๐ {๐, ๐} are ๐โ โ ๐๐ โ ๐๐๐๐ ๐๐ but not sg-closed. (iv) Consider the topology ๐ = {๐, ๐, {๐}, {๐}, {๐, ๐}}. Then the sets {๐}๐๐๐ {๐} are sg-closed but not ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐. (v) Consider the topology ๐ = {๐, ๐, {๐}}. Then the sets {๐}, {๐}, {๐}, {๐, ๐}๐๐๐ {๐, ๐} are ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐ but not ๐ผ โ ๐๐๐๐ ๐๐. (vi) Consider the topology ๐ = {๐, ๐, {๐}, {๐, ๐}}. Then the set {๐} is ๐ผ โ ๐๐๐๐ ๐๐ but not ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐. โ (vii) Consider the topology ๐ = {๐, ๐, {๐}}. Then the sets {๐} {๐, ๐} ๐๐๐ {๐, ๐} are ๐ โ ๐๐ โ ๐๐๐๐ ๐๐ but not preclosed. (viii) Consider the topology ๐ = {๐, ๐, {๐}, {๐, ๐}}. Then the set {๐} is pre-closed but not ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐. (ix) Consider the topology ๐ = {๐, ๐}. Then the sets {๐}, {๐}, {๐}, {๐, ๐}, {๐, ๐} ๐๐๐ {๐, ๐} are ๐โ โ ๐๐ โ ๐๐๐๐ ๐๐ but not gs-closed. (x) Consider the topology ๐ = {๐, ๐, {๐}, {๐, ๐, ๐}, {๐, ๐, ๐}}. Then the sets {๐} {๐, ๐} ๐๐๐ {๐, ๐} are ๐๐ โ ๐๐๐๐ ๐๐ but not ๐ โ โ ๐๐ โclosed. (xi) Consider the topology ๐ = {๐, ๐, {๐}, {๐}, {๐, ๐}}. Then the sets {๐}๐๐๐ {๐, ๐} are ๐๐ ๐ โ ๐๐๐๐ ๐๐ but not ๐ โ โ ๐๐ โclosed. (xii) Consider the topology ๐ = {๐, ๐, {๐}}. Then the set {๐} is ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐ but not gspclosed. (xiii) Consider the topology ๐ = {๐, ๐, {๐}}. Then the set {๐} is ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐ but not ๐ผgclosed. (xiv)Consider the topology ๐ = {๐, ๐, {๐}, {๐, ๐, ๐}, {๐, ๐, ๐}}. Then the set {๐}, {๐, ๐}, {๐, ๐} ๐๐๐ ๐ผ๐ โ ๐๐๐๐ ๐๐ but not ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐. (xv) Consider the topology ๐ = {๐, ๐, {๐}, {๐}, {๐, ๐}, {๐, ๐}}. Then the set {๐} is ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐ but not g๐ผ โ ๐๐๐๐ ๐๐. (xvi)Consider the topology ๐ = {๐, ๐, {๐}, {๐, ๐, ๐}, {๐, ๐, ๐}}. Then the set {๐}, {๐, ๐}๐๐๐ {๐, ๐} are ๐๐ผ โ ๐๐๐๐ ๐๐ but not ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐. Theorem: 3.9 For any two sets A and B ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)) = ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)) http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology [623] ISSN: 2277-9655 Impact Factor: 4.116 [Selvi* et al., 5(8): August, 2016] ICโข Value: 3.00 Proof: Since Aโ ๐ด โช ๐ต, we have ๐๐ โ (๐ ๐๐(๐ด)) โ ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)) and Since ๐ต โ ๐ด โช ๐ต, we have ๐๐ โ (๐ ๐๐(๐ต)) โ ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)). Therefore ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)) โ ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)). Also, ๐๐ โ (๐ ๐๐(๐ด)) and ๐๐ โ (๐ ๐๐(๐ต)) are the closed sets. Therefore ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)) is also a closed set. Again, Aโ ๐๐ โ (๐ ๐๐(๐ด)) and Bโ ๐๐ โ (๐ ๐๐(๐ต)), Implies โช ๐ต โ ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)). Thus, ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)) is a closed set containing ๐ด โช ๐ต. Since ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)) is the smallest closed set containing ๐ด โช ๐ต. We have ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)) โ ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)). Thus, ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)) = ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)) Theorem: 3.10 Union of two ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ sets in X is a ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ ๐ ๐๐ก ๐๐ ๐. Proof: Let A and B be two ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ ๐ ๐๐ก๐ . Let ๐ด โช ๐ต โ ๐บ, where G is ๐ โ โ ๐๐๐๐. Since A and B are ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐ ๐ ๐๐ก๐ , ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)) โ ๐บ. But by theorem 3.9 ๐๐ โ (๐ ๐๐(๐ด)) โช ๐๐ โ (๐ ๐๐(๐ต)) = ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)). Therefore ๐๐ โ (๐ ๐๐(๐ด โช ๐ต)) โ ๐บ. Hence ๐ด โช ๐ต is a ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ set. Theorem: 3.11 A subset A of X is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ if and only if ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด contains no non-empty ๐ โ โ ๐๐๐๐ ๐๐ set in X. Proof: Let A be a ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ set. Suppose that F is a non-empty ๐ โ โ ๐๐๐๐ ๐๐ subset of ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด. Now, F โ ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด. Then F โ ๐๐ โ (๐ ๐๐(๐ด)) โฉ ๐ด๐ , Since ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด = ๐๐ โ (๐ ๐๐(๐ด)) โฉ ๐ด๐ . Therefore F โ ๐๐ โ (๐ ๐๐(๐ด)) and F โ ๐ด๐ . Since ๐น ๐ is a ๐ โ -open set and A is a ๐ โ โ ๐๐ ๐๐๐๐ ๐๐, ๐๐ โ (๐ ๐๐(๐ด)) โ ๐น ๐ . That is ๐น โ ๐๐ โ (๐ ๐๐(๐ด)) โฉ [๐๐ โ (๐ ๐๐(๐ด))]๐ = ๐. That is ๐น = ๐, a contradiction. Thus, ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด contain no non-empty ๐ โ โ ๐๐๐๐ ๐๐ set in X. Conversely, assume that ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด contains no non-empty ๐ โ โ ๐๐๐๐ ๐๐ set. Let ๐ด โ ๐บ, G is ๐ โ โ ๐๐๐๐. Suppose that ๐๐ โ (๐ ๐๐(๐ด)) is not contained in G. then ๐๐ โ (๐ ๐๐(๐ด)) โฉ ๐บ ๐ is a non-empty ๐ โ โ ๐๐๐๐ ๐๐ set of ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด which is a contradiction. Therefore, ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด โ ๐บ and hence A is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐. Corollary: 3.12 A subset A of X is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ if and only ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด contains no non-empty closed set in X. Proof: The proof follows from the theorem 3.11 and the fact that every closed set is ๐ โ โ ๐๐๐๐ ๐๐ set in X. Corollary: 3.13 A subset A of X is ๐ โ โ ๐๐ โ ๐๐๐๐ ๐๐ if and only if ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด contains no non-empty open set in X. Proof: The proof follows from the theorem 3.11 and the fact that every open set is ๐ โ โ ๐๐๐๐ set in X. Theorem: 3.14 If a subset A of X is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ and ๐ด โ ๐ต โ ๐๐ โ (๐ ๐๐(๐ด)), then B is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ set in X. Proof: Let A be a ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ set such that ๐ด โ ๐ต โ ๐๐ โ (๐ ๐๐(๐ด)). Let ๐ be a ๐ โ โ ๐๐๐๐ set of X such that Bโ ๐. Since A is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐, we have ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ . Now, ๐๐ โ (๐ ๐๐(๐ด)) โ ๐๐ โ (๐ ๐๐(๐ต)) โ ๐๐ โ (๐๐ โ (๐ ๐๐(๐ด))) = ๐๐ โ (๐ ๐๐(๐ด)) โ ๐. That is ๐๐ โ (๐ ๐๐(๐ต)) โ ๐, U is ๐ โ โ ๐๐๐๐. Therefore B is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ set in X. The converse of the above theorem need not be true as seen from the following example. Example: 3.15 Consider the topological space ๐ = {๐, ๐, ๐} with topology ๐ = {๐, ๐, {๐}, {๐, ๐}}. Then A and B are ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ sets in (๐, ๐). But ๐ด โ ๐ต is not a subset of ๐๐ โ (๐ ๐๐(๐ด)). Theorem: 3.16 Let A be a ๐ โ โ ๐๐ ๐๐๐๐ ๐๐ in (๐, ๐). Then A is ๐ โ ๐๐๐๐ ๐๐ if and only if ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด is ๐ โ โ ๐๐๐๐. Proof: http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology [624] ISSN: 2277-9655 Impact Factor: 4.116 [Selvi* et al., 5(8): August, 2016] ICโข Value: 3.00 Suppose A is g-closed in X. Then, ๐๐ โ (๐ ๐๐(๐ด)) = ๐ด and so ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด = ๐ which is ๐ โ โ ๐๐๐๐ in X. Conversely, suppose ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด is ๐ โ โ ๐๐๐๐ in X. Since A is ๐ โ โ ๐๐ ๐๐๐๐ ๐๐, by the theorem 3.11, ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด contains no non-empty ๐ โ โ ๐๐๐๐ ๐๐ set in X. Then, ๐๐ โ (๐ ๐๐(๐ด)) โ ๐ด = ๐. Hence, A is g-closed. Remark 3.17 From the above discussion, we obtain the following implications. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] D.Andrijevic, Semi-preopen sets, Mat.Vesnik, 38 (1986), 24-32. S.P.Arya and T.Nour, Characterizations of s-normal spaces, Indian J.Pure Appl.Math., 21(1990), 717-719. P.Bhattacharyya and B.K.Lahiri, Semi generalized closed sets in topology, Indina J.Math., 29(1987), 375382 J.Dontchev, On generalizing semipreopen sets,Mem, Fac. 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