τ** - gs - Continuous Maps in Topological Spaces

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 10, Issue 6 Ver. VI (Nov - Dec. 2014), PP 25-28 www.iosrjournals.org τ** - gs - Continuous Maps in Topological Spaces T. Indira1 S.Geetha2 1 PG and Research Department of Mathematics, Seethalakshmi Ramaswami College, Trichy- 620 002, Tamil Nadu, India. 2 Department of Mathematics, Seethalakshmi Ramaswami College, Trichy- 620 002, Tamil Nadu, India. Abstract: In this paper, we introduce a new class of maps called τ **- generalized semi continuous maps in topological spaces and study some of its properties and relationship with some existing mappings. Key Words: scl*, τ** -topology, τ**-gs-open set, τ**-gs-closed set, τ**- gs-continuous maps. I. Introduction In 1970, Levine[7] introduced the concept of generalized closed sets as a generalization of closed sets in topological spaces. Using generalized closed sets, Dunham [3] introduced the concept of the closure operator cl* and a topology * and studied some of its properties. Pushpalatha, Easwaran and Rajarubi [10] introduced and studied *-generalized closed sets, and *-generalized open sets. Using *generalized closed sets, Easwaran and Pushpalatha [4] introduced and studied *-generalized continuous maps. The purpose of this paper is to introduce and study the concept of a new class of maps, namely ** -gs-continuous maps. 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, cl(A), scl(A), scl*(A) and AC denote the closure, semi-closure, generalized semi closure and complement of A respectively. II. Preliminaries Definition: 2.1 For the subset A of a topological space X, the generalized semi closure of A (i.e., scl*(A)) is defined as the intersection of all gs-closed sets containing A. Definition: 2.2 For the subset A of a topological space X, the topology [6] ** = {G : scl*(GC) = GC }. Definition: 2.3 A subset A of a topological space X is called **-generalized semi closed set [6] (briefly **-gs-closed) if scl*(A) G whenever A G and G is **-open. The complement of **-generalized semi closed set is called the **- generalized semi open set (briefly **- open). Definition: 2.4 The **- generalized semi closure operator cl ** for a subset A of a topological space (X, **) is defined by the intersection of all **- generalized semi closed sets containing A. (i.e.,) cl **(A) = ∩{G: A G and G is ** -gs-closed} Definition: 2.5 A map f: X → Y from a topological space X into a topological space Y is called: 1) continuous if the inverse image of every closed set (or open set) in Y is closed(or open) in X. (2) strongly gs-continuous if the inverse image of each gs-open set of Y is open in X. (3) semi continuous [12] if the inverse image of each closed set of Y is semi-closed in X. (4) sg-continuous [12] if the inverse image of each closed set of Y is sg-closed in X. (5) gs-continuous [13] if the inverse image of each closed set of Y is gs-closed in X. (6) gsp-continuous [2] if the inverse image of each closed set of Y is gsp-closed in x. (7) αg-continuous [5] if the inverse image of each closed set of Y is αg –closed in X. (8) pre-continuous [9] if the inverse image of each open set of Y is pre-open in X. (9) α-continuous [10] if the inverse image of each open set of Y is α-open in X. DOI: 10.9790/5728-10662528 www.iosrjournals.org 25 | Page τ** - gs - continuous maps in topological spaces (10) sp-continuous [1] if the inverse image of each open set of Y is semi-preopen in X. Remark: 2.6 1) In [6] it has been proved that every closed set is ** - gs closed. 2) In [6] it has been proved that every gs-closed set in X is ** -gs closed. III. ** - gs- Continuous Maps In Topological Spaces In this section, we introduce a new class of map namely **-generalized semi continuous map in topological spaces and study some of its properties and relationship with some existing mappings. Definition: 3.1 A map f:X →Y from a topological space X into a topological space Y is called ** -generalized continuous map(briefly ** - gs-continuous) if the inverse image of every closed set in Y is ** -gs-closed in X. Theorem: 3.2 Let f : X →Y be a map from a topological space (X, **) into a topological space (Y, **). The following statements are equivalent: a) f is ** - gs-continuous. b) the inverse image of each open set in Y is ** - gs-open in X. (ii) If f : X →Y is ** - gs-continuous, then f(cl **(A)) cl(f(A)) for every subset A of X. Proof: Assume that f : X→ Y is ** - gs-continuous. Let G be open in Y. Then GC is closed in Y. Since f is ** - gscontinuous, f-1(GC) is ** - gs-closed in X. But f-1(GC) = X – f-1(G). Thus X – f-1(G) is ** - gs-closed in X. Therefore (a) (b). Conversely, assume that the inverse image of each open set in Y is ** - gs-open in X. Let F be any closed set in Y. Then FC is open in Y. By assumption, f-1(FC) is ** - gs-open in X. But f-1(FC) = X - f-1(F). Therefore, X - f-1(F) is ** - gs-open in X and so f-1(F) is ** - gs-closed in X. Therefore, f is ** gs-continuous. Hence (b) (a). Thus (a) and (b) are equivalent. (ii) Assume that f is ** - gs-continuous. Let A be any subset of X, f(A) is a subset of Y. Then cl(f(A))) is a closed subset of Y. Since f is ** - gs-continuous, f-1(cl(f(A))) is ** - gs-closed in X and it containing A. But cl **(A) is the intersection of all ** - gs-closed sets containing A. cl ** (A) f-1(cl(f(A))) f(cl **(A)) cl(f(A)). Theorem: 3.3 If a map f : X →Y from a topological space X into a topological space Y is ** - gs-continuous then f is not gs-continuous. Proof: Let V be a closed set in Y. Then f-1(V) is ** - gs-closed in X, since f is ** - gs-continuous. But every ** - gs-closed set is not gs-closed. Therefore, f-1(V) is not gs-closed in X. Hence f is not gs-continuous. Theorem: 3.4 If a map f : X →Y from a topological space X into a topological space Y is continuous then it is ** - gscontinuous but not conversely. Proof: Let f : X →Y be continuous. Let V be a closed set in Y. Since f is continuous, f-1(V) is closed in X. By Remark: 2.6(2), f-1(V) is ** - gs-closed. Thus, f is ** - gs-continuous. The converse of the theorem need not be true as seen from the following example. Example: Let X =Y = {a,b,c}, = {X,Ф,{c}} and � = {Y,Ф, {b},{c},{b,c}}. DOI: 10.9790/5728-10662528 www.iosrjournals.org 26 | Page τ** - gs - continuous maps in topological spaces Let f :X →Y be an identity map. Then f is ** - gs-continuous. But f is not continuous. Since for the closed set V = {a} in Y, f-1(V) = {a} is not closed in X. Theorem: 3.5 If a map f : X →Y from a topological space X into a topological space Y is gs-continuous then it is ** - gs-continuous but not conversely. Proof: Let f : X →Y be gs-continuous. Let V be a closed set in Y. Since f is gs-continuous, f-1(V) is gs-closed in X. Also, by Remark: 2.6(2), f-1(V) is ** - gs-closed. Then, f is ** - gs-continuous. The converse of the theorem need not be true as seen from the following example. Example: Let X = Y = {a,b,c}, = {X,Ф,{a},{a,b}} and � = {Y,Ф, {a},{c},{a,c}}. Let f : X →Y be an identity map. Then f is ** - gs-continuous. But it is not gs-continuous. Since for the closed set V = {a,b} in Y, f-1(V) = {a,b} is not gs-closed in X and V = {b} in Y, f-1(V) = {b} is not gs-closed in X. Theorem: 3.7 If a map f : X→Y from a topological space X into a topological space Y is strongly gs-continuous then it is ** - gs-continuous but not conversely. Proof: Let f : X →Y be strongly gs-continuous. Let V be a closed set in Y, then G is gs-closed. Hence GC is gs-open in Y. Since f is strongly gs-continuous f-1(GC) is open in X. But f-1(Gc) = X – f-1(G). Therefore f-1(V) is closed in X. By Remark: 2.6(1), f-1(G) is ** - gs-closed in X. Therefore f is ** -gs-continuous. Note: If f : X → Y and g : Y → Z are both ** -gs-continuous then the composition gof ; x → z is not ** -gs-continuous mapping. Example: Let X = Y = Z = {a,b,c}, = {X,Ф,{a},{b},{a,b}}and� = {Y,Ф, {a}}. Define f : X → Y by f(a) = c, f(b) = a and f(c) = b and define g : Y → Z by g(a) = a, g(b) = c, g(c) = b. Then f and g are ** -gs-continuous mappings. The set {b} is closed in Z. (gof)-1 ({b}) = f-1(g-1({b})) = f-1({c}) = {a} which is not ** -gs-closed in X. Hence gof is not ** -gs-continuous. Remark: 3.8 From the above discussion, we obtain the following implications. Let X = Y = {a,b,c}. Let f : X →Y be an identity map. 1) Let = {X,Ф,{a},{a,c}} and � = {Y,Ф,{a},{a,b},{a,c}}. Then f is both ** - gs-continuous and semicontinuous. 2) Let = {X,Ф,{c}} and � = {Y,Ф,{b},{c},{b,c}}. Then f is ** - gs-continuous. But it is not semicontinuous. Since for the closed set V = {a,c} in Y, f-1(V) = {a,c} is not semi-closed in X. 3) Let = {X,Ф,{a},{b},{a,b},{a,c}} and � = {Y,Ф, {a},{a,c}}. Then f is both ** - gs-continuous and sgcontinuous. 4) Let = {X,Ф,{a}} and � = {Y,Ф,{b},{c},{b,c},{a,c}}. Then f is ** - gs-continuous. But it is not gscontinuous. Since for the closed set V = {a} in Y, f-1(V) = {a} is not gs- closed in X. 5) Let = {X,Ф,{c}} and � = {Y,Ф,{a},{b}{b,c},{a,b}}. Then f is ** - gs-continuous. But it is not gspcontinuous. Since for the closed set V = {c} in Y, f-1(V) = {c} is not gsp-closed in X. 6) Let = {X,Ф,{a,c}} and � = {Y,Ф,{a},{c},{a,b},{a,c}}. Then f is gsp-continuous. But it is not ** - gscontinuous. Since for closed set V = {c} in Y, f-1(V) = {c} is not ** - gs-closed in X. 7) Let = {X,Ф,{b}} and � = {Y,Ф,{c},{a,c}}. Then f is ** - gs-continuous. But it is not αg-continuous. Since for the closed set V = {b} in Y, f-1(V) = {b} is not αg-closed in X. 8) Let = {X,Ф,{a}} and � = {Y,Ф,{c},{a,c},{b,c}}. Then f is ** - gs-continuous. But it is not precontinuous. Since for the closed set V = {b,c} in Y, f-1(V) = {b,c} is not pre-open in X. DOI: 10.9790/5728-10662528 www.iosrjournals.org 27 | Page τ** - gs - continuous maps in topological spaces 9) Let = {X,Ф,{a},{b,c}} and � = {Y,Ф,{a},{c},{a,c},{b,c}}. Then f is ** - gs- continuous. But it is not α-continuous. Since for the closed set V = {a,c} in Y,f-1(V) = {a,c} is not α-open in X. 10) Let = {X,Ф,{a},{a,c} and � = {Y,Ф,{a},{a,b},{a,c}}. Then f is both sp-continuous and ** -gscontinuous. 11) Let = {X,Ф,{c}} and � = {Y,Ф,{b},{c},{a,c},{b,c}}. Then f is ** -gs-continuous. But it is not spcontinuous. Since for the open set V= {b} in Y,f-1(V) = {b} is not sp-open in X. IV. Conclusion The class of ** -gs-continuous maps defined using ** -gs-closed sets. The ** -gs-closed sets can be used to derive a new homeomorphism, connectedness, compactness and new separation axioms. This concept can be extended to bitopological and fuzzy topological spaces. References [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. [10]. [11]. [12]. [13]. M.E. Abd El-Monsef, S.N.El.DEEb and R.A.Mahmoud, β-open sets and and β-continuous mappings, Bull. Fac. Sci. Assiut Univ.12(1)(1983),77-90. J.Dontchev, On generalizing semipreopen sets, Mem. Fac. Sci. Kochi Uni. Ser A, Math., 16(1995), 35-48. W. Dunham, A new closure operator for non-T1 topologies, Kyungpook Math.J.22(1982), 55-60. S. Easwaran and A Pushpalatha, * - generalized continuous maps in topological spaces, International J. of Math Sci & Engg. Appls.(IJMSEA) ISSN 0973-9424 Vol.3, No.IV,(2009),pp.67-76. Y. Gnanambal, On generalized preregular sets in topological spaces, Indian J. Pure Appl. Math.(28)3(1997),351-360. T. Indira and S. Geetha , ** - gs- closed sets in topological spaces, Antarctica J. Math.,10(6)(2013),561-567. N. Levine, Generalized closed sets in topology, Rend. Circ. Mat..Palermo,19,(2)(1970), 89- 96. A.S.Mashhour, I.A.Hasanein and S.N.El-Deeb, On precontinuous and weak precontinuous functions, Proc. Math. Phys. Soc.Egypt 53(1982),47-53. A.S. Mashhour, I.A.Hasanein and S.N.El-Deeb, τn α-continuous and α-open mappings, Acta. Math.Hunga.41(1983),213-218. A.Pushpalatha, S.Easwaran and P.Rajarubi, *-generalized closed sets in topological spaces, Prodeedings of World Congress on Engineering 2009 Vol II WCE 2009, July 1-3,2009, London, U.K., 1115-1117. P.Sundarm, H.Maki and K.Balachandran, sg-closed sets and semi-T1/2 spaces. Bull.Fukuoka Univ. Ed..Part III,40(1991),33-40. Semi open sets and semi continuity in topological spaces, Amer. Math., Semigeneralized closed and generalized closed maps, Mem.Fac.Sci.Kochi . DOI: 10.9790/5728-10662528 www.iosrjournals.org 28 | Page