On Generalized gp*- Closed Map. in Topological Spaces

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1 Applied Mathematical Sciences, Vol. 8, 2014, no. 9, HIKARI Ltd, On Generalized gp*- Closed Map in Topological Spaces S. Sekar Department of Mathematics Government Arts College (Autonomous) Salem , Tamil Nadu, India P. Jayakumar Department of Mathematics Paavai Engineering College, Pachal, Namakkal , Tamil Nadu, India Copyright 2014 S. Sekar and P. Jayakumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, the authors introduce a new class of generalized gp*-closed map in topological spaces and study some of their properties as well as inter relationship with other closed map Mathematics Subject Classification: 54C05, 54C08, 54C10 Keywords: gp*-closed map and gp*-open map 1 Introduction Different types of Closed and open mappings were studied by various researchers. In 1998, T.Noiri introduced new type of set called gp-closed set. A.A.Omari and M.S.M. Noorani introduced and studied b-closed map. The aim of this paper is to introduce gp* closed map and to continue the study of its relationship with various generalized closed maps. Throughout this paper (X,τ ) and (Y,σ ) represent the non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned.

2 416 S. Sekar and P. Jayakumar 2 Preliminaries Definition 2.1[4]: Let A subset A of a topological space (X,τ ), is called a pre-open set if A Int(cl (A)). Definition 2.2 [1]: Let A subset A of a topological space (X,τ ), is called a generalized closed set (briefly g-closed) if cl (A) U whenever A U and U is open in X. Definition 2.3 [5]: Let A subset A of a topological space (X,τ ), is called a generalized pre- closed set (briefly gp- closed) if pcl (A) U whenever A U and U is open in X. Definition 2.4 [3]: Let A subset A of a topological space (X,τ ), is called a generalized pre-closed set (briefly pg-closed) if pcl(a) U whenever A U and U is pre-open in X. Definition 2.5 [8]: Let A subset A of a topological space (X,τ ), is called a generalized pre- closed set (briefly g*- closed) if cl (A) U whenever A U and U is g-open in X. Definition 2.6 [10]: Let A subset A of a topological space (X,τ ), is called a generalized pre- closed set (briefly g*p-closed) if pcl (A) U whenever A U and U is g-open in X. Definition 2.7 [7]: Let A subset A of a topological space (X,τ ), is called a generalized pre- closed set (briefly strongly g- closed) if cl (A) U whenever A U and U is g-open in X. Definition 2.8 [6]: Let A subset A of a topological space (X,τ ), is called a generalized pre- closed set (briefly α g- closed) if α cl(a) U whenever A U and U is open in X. Definition 2.9 [9]: Let A subset A of a topological space (X,τ ), is called a generalized pre- closed set (briefly g# closed) if cl(a) U whenever A U and U is α g-open in X. Definition 2.10 [15]: A subset A of a topological space (X, τ ), is called gp * -closed set if cl (A) U whenever A U and U is gp open in X. 3 On regular generalized b-closed map In this section we introduce gp*-closed map(gp*-closed),gp*-open map (gp*-open) and study the some of their properties. Definition 3.1: Let X and Y be two topological spaces. A map f: (X,τ ) (Y,δ ) is called gp*-closed map(briefly gp*-closed), if the image of every closed set in X is gp*-closed in Y. Theorem 3.2. Every closed map is gp*-closed but not conversely.

3 On generalized gp*- closed map in topological spaces 417 Proof: Let f: (X,τ ) (Y,δ ) is closed map and V be an closed set in X then f (V) is closed in Y. Hence gp*-closed in Y. Then f is gp*-closed. The converse of above theorem need not be true as seen from the following example. Example 3.3 Consider X = Y = {a,b,c}, τ ={ X, ϕ, {b,c}} and σ ={Y,ϕ,{a},{c},{a,c},{b,c}}. Let f: (X,τ ) (Y, σ ) be defined by f(a) = a, f(b) = b, f(c) = c. The map is gp*-closed but not closed as the image of and {a,b} in X is {a,b} is not closed in Y. Theorem 3.4. Every g*-closed map is gp*-closed but not conversely. Proof: Let f: (X, τ ) (Y,δ ) is g*-closed map and V be an closed set in X then f (V) is closed in Y. Hence gp*-closed in Y. Then f is gp*-closed. Example 3.4 Consider X = Y = {a,b,c}, τ ={ X, ϕ,{b}} and σ ={Y,ϕ }. Let f: (X,τ ) (Y, σ ) be defined by f(a) = c, f(b) = a, f(c) = b. The map is gp*-closed but not g*closed as the image of and {a,c} in X is {b,c} is not g*-closed in Y. Theorem 3.5 Every pg-closed map is gp*-closed but not conversely. Proof: Let f: (X, τ ) (Y,δ ) be pg-closed map and V be an closed set in X then f (V) is closed in Y. Hence gp*-closed in Y. Then f is gp*-closed. Example 3.6 Consider X = Y = {a,b,c}, τ ={ X, ϕ, {a},{a,c}} and σ ={Y,ϕ,{a},{c},{a,c},{b,c}}. Let f: (X,τ ) (Y, σ ) be defined by f(a) = b, f(b) = a, f(c) = c. The map is gp*-closed but not pg closed as the image of {b,c} in X is {a,c} is not pg closed in Y. Theorem 3.7 Every g*p map is gp*-closed but not conversely. Proof: Let f: (X, τ ) (Y,δ ) be g*p map and V be an closed set in X then f (V) is closed in Y. Hence gp*-closed in Y. Then f is gp*-closed. Example 3.8 Consider X = Y = {a,b,c}, τ ={ X, ϕ, {a},{a,b}} and σ ={Y,ϕ,{b}}. Let f: (X,τ ) (Y, σ ) be defined by f(a) = b, f(b) = c, f(c) = a. The map is gp*-closed but not g*p-closed as the image of and {c,b} in X is {a,c} is not g*p-closed in Y. Theorem 3.9 Every gp*closed map is strongly g closed map but not conversely. Proof: Let f: (X, τ ) (Y,δ ) gp*- closed map and V be an closed set in X then f (V) is closed in Y. Hence strongly g closed in Y. Then f is strongly g closed.

4 418 S. Sekar and P. Jayakumar Example 3.10 Consider X = Y = {a,b,c}, τ ={ X, ϕ,{b},{a,b}} and σ ={Y,ϕ,{c},{b,c}}. Let f: (X,τ ) (Y, σ ) be defined by f(a) = c, f(b) = b, f(c) = a. The map is strongly g closed but not gp*closed as the image of and {a,c} in X is {a,c} is not gp*closed in Y. Theorem 3.11 Every gp*-closed map is g#-closed but not conversely. Proof: Let f: (X, τ ) (Y,δ ) be gp*-closed map and V be an closed set in X then f (V) is closed in Y. Hence g# closed in Y. Then f is g# closed in Y. Example 3.12 Consider X = Y = {a,b,c}, τ ={ X, ϕ,{a},{a,b}} and σ ={Y,ϕ,{a},{b},{a,b},{b,c}}. Let f: (X,τ ) (Y, σ ) be defined by f(a) = b, f(b) = c, f(c) = a. The map is gp*-closed but not gα -closed as the image of and {b,c} in X is {a,c} is not closed in Y. Theorem 3.13 Every gp*closed map is α g*closed but not conversely. Proof: Let f: (X, τ ) (Y,δ ) be gp* closed map and V be an closed set in X then f (V) is closed in Y. Hence α g*-closed in Y. Then f is α g*-closed. Example 3.14 Consider X = Y = {a,b,c}, τ ={ X,ϕ,{a}} and σ ={Y,ϕ,{b}}. Let f: (X,τ ) (Y, σ ) be defined by f(a) = b, f(b) = a, f(c) = c. The map is α g*closed but not gp* closed as the image of {a,c} in X is {b,c} is not closed in Y. 4 Application Theorem 4.1 A map f: (X,τ ) (Y,σ ) is continuous and gp*-closed set A is gp*-closed set of X then f(a) is gp* closed in Y. Proof: Let f(a) U where U is gp open set in Y. Since f is continuous, f -1 (U) is open set containing A. Hence cl(a) f -1 (U) (as A is gp*-closed). Since f is gp*-closed f (cl(a)) U is gp* closed set cl(f(cl(a)) U, Hence cl(a) U. So that f(a) is gp*-closed set in Y. Theorem 4.2. If a map f: (X, τ ) (Y,σ ) is continuous and closed set and A is gp*-closed then f(a) is gp*-closed in Y. Proof: Let F be a closed set of A then F is gp*-closed set. By theorem 4.8 f(a) is gp*-closed. Hence f A (F) = f(f) is gp*-closed set of Y. Here f A is gp*-closed and also continuous. Theorem 4.3 If f: (X,τ ) (Y,σ ) is closed map and g: (Y, σ ) (Z, η ) is gp*-closed map, then the composition g f : (X, τ ) (Z, η ) is gp*-closed map. Proof: Let F be any closed set in (X, τ ). Since f is closed map, f(f) is closed set in (Y,σ ). Since g is gp*-closed map, g(f(f)) is gp*-closed set in (Z, η ). That is g.f(f) = g(f(f)) is gp* closed. Hence g f is gp* closed map.

5 On generalized gp*- closed map in topological spaces 419 Remark: If f: (X, τ ) (Y,σ ) is gp*-closed map and g: (Y, σ ) (Z, η ) is closed map, then the composition need not gp*-closed map. Theorem 4.4 A map f: (X, τ ) (Y,σ ) is gp*-closed if and only if for each subset S of (Y,σ ) and each open set U containing f -1 (S) U, there is a gp*-open set V of (Y,σ ) such that S V and f -1 (V) U. Proof: Suppose f is gp*-closed. Let S Y and U be an open set of (X,τ ) such that f -1 (S) U. Now X-U is closed set in (X,τ ). Since f is gp*-closed, f(x-u) is gp*-closed set in (Y,σ ). There fore V = Y f(x-u) is an gp*-open set in (Y,σ ). Now f -1 (S) U implies S V and f -1 (V) = X - f -1 (f(x-u)) X-(X-V) = U. (ie) f -1 (V) U. Conversely, Let F be a closed set of (X,τ ). Then f -1 (f(f c )) F c. and Fc. is an open in(x, τ ). By hypothesis, A gp*-open set V in (Y,σ ) such that f(f c ) V and f -1 (V) F c F f -1 (V) c. Hence V c f(f) f(((f -1 (V)) c ) c V c f(v) V c. Since V c is gp*-closed, f(f) is gp*-closed. (ie) f(f) is gp*-closed in Y and. Therefore f is gp*-closed. Theorem 4.5 If f: X 1 X 2 Y 1 Y 2 is defined as f(x 1,x 2 ) = (f 1 (x 1 ),f 2 (x 2 )), then f: X 1 X 2 Y 1 Y 2 is gp* closed map. Proof: Let U 1 U 2 X 1 X 2 where U i gp*cl(x i ), for i = 1,2. Then f(u 1 U 2 ) = f 1 (U 1 ) f 2 (U 2 ) gp*cl(x 1 Y 2 ). Hence f is gp*-closed set. Theorem 4.6 Let h: X X 1 X 2 be gp*-closed map and Let f i : X X i be define as h(x) = (x 1,x 2 ) and f i (x) = x i, then f i : X X i is gp*-closed map for i = 1,2. Proof: Let U 1 U 2 X 1 X 2, then f 1 (U 1 ) = h 1 (U 1 X 2 ) gp*cl(x), there fore f 1 is gp*-closed. Similarly we have f 2 is gp*-closed. Thus f i is gp*-closed map for i=1, 2. Theorem 4.7 For any bijection map f: (X,τ ) (Y,σ ), the following statements are equivalent: (i) f -1 : (Y,σ ) (X, τ ) is gp*-continuous. (ii) f is gp*-open map (iii) f is gp*-closed map. Proof: (i) (ii) Let U be an open set of (X, τ ). Bu assumption, (f -1 ) -1 (U) = f(u) is gp*-open in (Y,σ ) and so f is gp*-open. (ii) (iii) Let F be a closed set of (X, τ ). Then F c is open set in (X,τ ). By assumption f(f c ) is gp*-open in (Y,σ ). Therefore f (F c ) = f (F) c is gp*-open in (Y,σ ). That is f(f) is gp*-closed in (Y,σ ). Hence f is gp*-closed. (iii) (i) Let F be a closed set of (X, τ ). By assumption, f (F) is gp*-closed in (Y,σ ). But f (F) = (f -1 ) -1 (F) (f -1 ) is continuous.

6 420 S. Sekar and P. Jayakumar 5. On regular generalized b-open map Definition 5.1: Let X and Y be two topological spaces. A map f: (X,τ ) (Y,δ ) is called regular generalized b-open(briefly gp*-open) if the image of every open set in X is gp*-open in Y. Theorem 5.2 Every open map is gp*-open but not conversely. Proof: Let f: (X,τ ) (Y,δ ) is open map and V be an open set in X then f(v) is open in Y. Hence gp*-open in Y. Then f is gp*-open. The converse of above theorem need not be true as seen from the following example. Example 5.3 Consider X = Y = {a,b,c}, τ ={X,ϕ,{a}} and σ ={Y,ϕ,{b}}. Let f: (X,τ ) (Y, σ ) be defined by f(a) = b, f(b) = c, f(c) = a. The map is gp*-open but not open as the image of and {b,c} in X is {a,c} is not open in Y. Theorem 5.4 A map f: (X, τ ) (Y,σ ) is gp*-closed set if and only if for each subset S of Y and for each open set U containing f -1 (S) U there is a gp*-open set V of Y such that S U and f -1 (V) U. Proof: Suppose f is gp*-closed set. Let S Y and U be an open set of (X,τ ) such that f -1 (S) U. Now X-U is closed set in (X, τ ). Since f is gp* closed, f(x-u) is gp* closed set in(y,σ ). Then V = Y- f(x-u) is gp* open set in (Y, σ ). Therefore f -1 (S) U implies S V and f -1 (V) = X- f -1 (f(x-u)) X-(X-V) = U. (ie) f -1 (V) U. Conversely, Let F be a closed set of (X, τ ). Then f -1 (f (F c )) Fc and F c is an open in (X,τ ). By hypothesis, a gp* open set V in (Y,σ ) such that f(fc ) V and f -1 (V) F c F (f -1 (V) c. Hence V c f(f) f((( f-1 (V)) c ) c ) V c f(v) V c. Since V c gp*-closed, f(f) is gp*-closed. (ie) f(f) is gp*-closed in (Y,σ ) and therefore f is gp*-closed. Theorem 5.5 For any bijection map f: (X,τ ) (Y,σ ), the following statements are equivalent. (i) f -1 : (X,τ ) (Y,σ ) is gp*-continuous. (ii) f is gp* open map. (iii) f is gp*-closed map. Proof: (i) (ii) Let U be an open set of (X,τ ). By assumption (f -1 ) -1 (U) = f(u) is gp*-open in (Y,σ ). Therefore f is gp*-open map. (ii) (iii) Let F be closed set of (X,τ ), Then F c is open set in (X,τ ). By assumption, f(f c ) is gp*-open in (Y,σ ). Therefore f(f) is gp*-closed in (Y,σ ). Hence f is gp*-closed. (iii) (i) Let F be a closed set of (X,τ ), By assumption f(f) is gp*-closed in

7 On generalized gp*- closed map in topological spaces 421 (Y,σ ). But f(f) = (f -1 ) -1 (F). Hence f -1 : (X,τ ) (Y,σ ) is gp*-continuous. ACKNOWLEDGMENTS The authors gratefully acknowledge the Dr. A. R. Rajamani, Principal, Government Arts College (Autonomous), Salem , for encouragement and support. The authors also thank Mr. K. Mariappa, Assistant Professor, Department of Mathematics, King College of Technology, Namakkal , Tamil Nadu, India, for his kind help in basic ideas of this research paper and encouragement. REFERENCES [1] Ahmad Al-Omari and Mohd. Salmi Md. Noorani, On Generalized b-closed sets. Bull. Malays. Math. Sci. Soc(2) 32(1) (2009), [2] D. Andrijevic, Semi-pre open sets mat.vesnik 38 (1) 1986, [3] D. Andrijevic, b-open sets, Mat.Vesink, 48 (1996), [4] S.P. Arya and T. Nour, Characterizations of s-normal spaces, Indian J. Pure Applied Maths 21 (8) (1990) [5] P. Bhattacharya and B.K. Lahiri, Semi-generalized closed sets in Topology, Indian J. Math., 29(1987), [6] Y. Gnanambal, On gpr continuous functions in topological spaces, Indian J. Pure Applied Math 30(6) (1999), [7] D. Iyappan and N. Nagaveni, On Semi Generalized b-continuous Maps, Semi Generalized b-closed maps in Topological Space, Int. Journal of Math. Analysis, Vol. 6, 2012, no. 26, [8] N. Levine, Generalized closed sets in topology, Tend Circ.,Mat. Palermo (2)19 (1970), [9] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer.Math.Monthly, 70(1963), [10] H. Maki, R. Devi and K. Balachandran, Associated topologies of generalized α - closed Sets and α -generalized closed sets Mem. Fac. Sci. Kochi. Univ. Ser. A. Math. 15 (1994),

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