ScienceDirect. -IRRESOLUTE MAPS IN TOPOLOGICAL SPACES K. Kannan a, N. Nagaveni b And S. Saranya c a & c

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1 Available online at ScienceDirect Procedia Computer Science 47 (2015 ) ON βˆ g -CONTINUOUS AND βˆ g -IRRESOLUTE MAPS IN TOPOLOGICAL SPACES K. Kannan a, N. Nagaveni b And S. Saranya c a & c Department of Mathematics, Hindusthan College of Engineering & Technology, Coimbatore , India. b Department of Mathematics, Coimbatore Institute of Technology, Coimbatore , Tamil Nadu, India. Abstract We investigated a new class of -continuous maps and -irresolute maps in topological spaces and study some of its basic properties and relations among them 2015 Published The Authors. by Elsevier Published B.V. by This Elsevier is an B.V. open access article under the CC BY-NC-ND license Peer-review ( under responsibility of organizing committee of the Graph Algorithms, High Performance Implementations and Peer-review Applications under (ICGHIA2014). responsibility of organizing committee of the Graph Algorithms, High Performance Implementations and Applications (ICGHIA2014) Keywords: -continuous maps and -irresolute maps 1. Introduction Levine[11] introduced the concepts of semi-open sets and semi-continuous in a topological space and investigated some of their properties. Strong forms of stronger and weaker forms of continuous map have been introduced and investigated by several mathematicians. Ekici [8] introduced and studied b-continuous functions in topological spaces. Omari and Noorani [16] introduced and studied the concept of generalized g-closed sets and g -continuous maps in topological spaces. Kannan and Nagaveni [10] introduced and studied the properties of βˆ -generalized closed sets and open sets in topological spaces. Crossley and Hildebrand [6] introduced and investigated irresolute functions which are stronger than semi continuous maps but are independent of continuous maps. Di Maio and Noiri[7], Faro[9],Cammaroto and Noiri [5], Maheswari and Prasad[13] and Sundaram [17] have introduced and studied quasi-irresolute and strongly irresolute maps strongly α-irresolute maps, almost irresolute maps, α-irresolute maps and gcirresolute maps are respectively. *K.Kannan kkmaths2010@gmail.com Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of organizing committee of the Graph Algorithms, High Performance Implementations and Applications (ICGHIA2014) doi: /j.procs

2 K. Kannan et al. / Procedia Computer Science 47 ( 2015 ) The aim of this paper is to introduce and study the concepts of new class of maps namely βˆ g - continuous maps and βˆ g -irresolute maps. Throughout this paper (X,τ) and (Y,σ)(or simply X and Y) represents the non-empty topological spaces on which no separation axiom are assumed, unless otherwise mentioned. For a subset A of X, cl(a) and int(a) represents the closure of A and interior of A respectively. 2. Preliminaries Let us recall the following definitions which are used in our sequel. Definition 2.1 [15]: A subset A of a topological space X, is called a (-open set if A( intclinta and a (-closed set if clintcla(a Definition 2.2 [2]: A subset A of a topological space X, is called a semi-preopen set if A ( clintcla and a semi-preclosed set if intclinta ( A Definition 2.3 [3]: A subset A of a topological space (X, τ ) is called a b-open set if int(cl(a)) and b-closed set if cl(int(a)) int(cl(a)) A. A cl(int(a)) Definition 2.4 [12]: 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 2.5 [11]: A subset A of a topological space (X, τ ) is called a semi-open set if A cl(int(a)) and semi closed set if int(cl(a)) A. Definition 2.6 [10]: A subset A of a topological space set) if clintcl(a) U whenever A U and U is open in X. X, is called -closed set (β-generalized closed Definition 2.7[4]: A function f : X Y is said to be generalized continuous (g- continuous) if f 1 (V ) is g-open in X for each open set V of Y. Definition 2.8 [8]: A function f: X Y is said to be b-continuous if for each x X and for each open set of V of Y containing f (x), there exists U bo(x, x) such that f (U ) V. Definition: 2.9 [14]: A function f: X Y is said to be α - continuous if f -1 (V) is α - open in X for each open set V of Y. Definition 2.10 [1]: A map f: X Y is semi pre- continuous if and only if for every closed set V of Y, f -1 (V)is semi pre-closed in X. Definition 2.10 [6]: A map f :(X,τ) (Y,σ) from a topological space X into a topological space Y is called irresolute if f 1 (V ) is semi-closed in X for every semi-closed set V of Y. 3. on -continuous maps in Topological Spaces Definition 3.1: A map f : (X,()((Y,() is said to be (Y,() is -closed in (X,(). -continuous if the inverse image of every closed set in

3 370 K. Kannan et al. / Procedia Computer Science 47 ( 2015 ) Theorem 3.2: Every continuous map is -continuous map, but not conversely. Proof: Let V be a closed set in (Y,().Then f -1 (V) is a closed set in (X,() as f is a continuous map. Since every closed set is -closed, f -1 (V) is -closed set in (X,(). Therefore f is a -continuous map. Example 3.3: Let X = Y = {a, b, c} with topologies τ = {, {a, b}, X} and b},y}. Let f :(X,()((Y,() be the identity map. Then the map f is inverse image of a closed set {b,c} in Y is {b,c} which is ( ={,{a},{a, -continuous but not continuous, as the -closed but not closed in X. Theorem 3.4: Every (-continuous map is -continuous map, but not conversely. Proof: Let V be a closed set in (Y,(). Then f -1 (V) is an (-closed set in (X,() as f is an (-continuous map. Since every (-closed set is βˆ g -closed, f -1 (V) is βˆ g - closed set in (X,(). Therefore f is a βˆ g -continuous map. Example 3.5: Let X = Y = {a, b, c} with topologies τ ={,{ b},{ b, c}, X} and ( ={,{ b},{c},{ b, c}, Y}. Let f :(X,()((Y,() be the identity map. Then the map f is a -continuous but not (- continuous, as the inverse image of a closed set {c} in Y is {a, b} which is -closed but not α-closed in X. Theorem 3.6: Every β-continuous map is -continuous map, but not conversely. Proof: Let V be a closed set in (Y,().Then f -1 (V) is a β -closed set in (X,() as f is a β-continuous map. Since every β-closed set is -closed, f -1 (V) is -closed set in (X,(). Therefore f is a -continuous map. Example 3.7: Let X = Y = {a, b, c} with topologies τ ={,{b},{b, c},x} and ( ={,{a},{a, b},y}. Let f : (X,()((Y,() be defined by f (a) = c, f (b) = b, f (c) = a. Then the map f is -continuous but not β-continuous, as the inverse image of a closed set {b, c} in Y is {a, b} which is -closed but not β - closed in X. Remark 3.8: The concept of -continuous map and b-continuous map are independent of each other as seen from the following example. Example 3.9: Let X = Y = {a, b, c} with topologies ( = {,{ b}, { b, c }, X} and ( ={,{c},y}. Let f : (X,()((Y,() be the identity map. Then the map f is -continuous but not b-continuous, as the inverse image of a closed set {a, b} in Y is {a, b} which is -closed but not b-closed in X. Example 3.10: Let X = Y = {a, b, c} with topologies ( ={,{a},{b},{a, b},x} and ( = {,{b, c},y }. Let f :( X,()((Y,() be the identity map. Then the map f is b-continuous but not βˆ g -continuous, as the

4 K. Kannan et al. / Procedia Computer Science 47 ( 2015 ) inverse image of a closed set {b, c} in Y is {a} which is b-closed but not -closed in X. Theorem 3.11: Every g-continuous map is -continuous map, but not conversely. Proof: Let V be a closed set in (Y,().Then f -1 (V) is a g-closed set in (X,() as f is a g-continuous map. Since every g-closed set is β^g -closed, f -1 (V) is -closed set in (X,(). Therefore f is a -continuous map. Example 3.12: Let X =Y ={a, b, c} with topologies ( ={,{a},{a, b}, X } and ( = {, {a, c}, {b},y}. Let f : (X,()((Y,() be the identity map. Then the map f is -continuous but not g-continuous as the inverse image of a closed set {a, c} in Y is {b} which is -closed but not g-closed in X. Remark 3.13: From the above discussions and the known result we have the following implications, Continuity (-continuity -continu b-continuity β continuity g-continuity Theorem 3.14: Let f : (X,()((Y,() be a map from a topological space X into a topological space Y. Then, (a) f is -continuous. (b)the inverse image of each open set in Y is -open in X. Proof : Assume that f : X(Y be -continuous. Let G be open in Y. Then G C is closed in Y. Since f is -continuous, f -1 (G C ) is βˆ g -closed in X. But f -1 (G C ) = X ( f -1 (G). Thus X( f -1 (G) is -closed in X and so f -1 (G) is -open in X. Therefore (a) implies (b). Conversely assume that the inverse image of each open set in Y is -open in X. Let F be any closed set in Y. Then F C is open in Y. By assumption, f -1 (F C ) is -open in X. But f -1 (F C ) =X( f -1 (F). Thus X( f -1 (F) is -open in X and so f -1 (F) is -closed in X. Therefore f is -continuous. Hence (b) implies (a). Thus (a) and (b) are equivalent. Theorem 3.15: If f : (X,()((Y,() and g :(Y,()((Z,() are any two functions, then the composition g f : (X,()((Z,() is -continuous if g is continuous and f is -continuous. Proof: Let V be any closed set in (Z,(). Since g is continuous, g -1 (V) is closed set in (Y,() and since f is -continuous, f -1 ( g -1 (V)) is -closed set in (X,(). That is (g f ) -1 (V) is βˆ g -closed in X. Thus g f is -continuous.

5 372 K. Kannan et al. / Procedia Computer Science 47 ( 2015 ) Remark 3.16: The composition of two βˆ g seen from the following example. -continuous maps need not be -continuous as Example 3.17: Let X =Y = Z = {a, b, c} with topologies ( ={,{c}, X }, ( = {, {b},{b,c},y } and (={,{a,b},z}. Define a map f : (X,()( (Y,() be the identity map and also g : (Y,()((Z, () be the identity map. Then both f and g are -continuous, but their composition g f is not -continuous, because {c} is a closed set of (Z,(). Therefore (g f) -1 ({c})={c}is not a -closed set of (X,(). Hence g f is not -continuous. 4. on -irresolute maps in Topological Spaces Definition 4.1: A map f :(X,()((Y,() is said to be in (Y,() is -closed in (X,(). -irresolute if the inverse image of every -closed set Theorem 4.2: A map f :(X,()((Y,() is in (Y,() is -open in (X,(). -irresolute if and only if the inverse image of every -open set Proof: Assume that f is -irresolute. Let A be any -open set in Y. Then A C is -closed set in Y. Since f is -irresolute, f -1 (A C ) is -closed set in X. But f -1 (A C ) = X( f -1 (A) and so f -1 (A) is a -open set in X. Hence the inverse image of every -open set in Y is -open in X. Conversely assume that the inverse image of every -open set in Y is - open in X. Let A be any - closed set in Y. Then A C is -open in Y. By assumption, f -1 (A C ) is -open in X. But f -1 (A C ) = X( f - 1 (A) and so f -1 (A) is -closed in X. Therefore f is -irresolute. Theorem 4.3: If a map f :(X,()((Y,() is -irresolute, then it is - continuous but not conversely. Proof: Assume that f is -irresolute. Let F be any closed set in Y. Since every closed set is -closed, F is -closed in Y. Since f is -irresolute, f -1 (F) is -closed in X. Therefore f is -continuous. Remark 4.4: The converse of the above theorem need not be true as seen from the following example. Example 4.5:Let X =Y ={a, b, c} with topologies (={,{b},{c},{b, c},x} and ( ={,{b},{b,c},y}. Let f : (X,()((Y,() be the identity map. Then the map f is continuous but not -

6 K. Kannan et al. / Procedia Computer Science 47 ( 2015 ) irresolute as {c} is βˆ g irresolute. -closed in Y but f -1 ({ c}) ={c} is not βˆ g Theorem 4.6: Let X, Y and Z be any topological spaces. For any any -continuous map g : (Y, ()((Z,(), the composition Proof: Let F be any closed set in Z. Since g is -continuous, g -1 (F) is irresolute, f -1 (g -1 (F)) is continuous. -closed in X. Therefore f is not βˆ g - - irresolute map f :(X,()((Y,() and g f : (X,()( (Z,() is -continuous. - closed set in Y. Since f is - -closed in X. But f -1 (g -1 (F))=(g f ) -1 (F). Therefore g f : (X,()( (Z,() is - Remark 4.7: The irresolute maps and following example. -irresolute maps are independent of each other as seen from the Example 4.8:Let X=Y={a, b, c}with topologies ( ={,{a},{b},{a, b},x} and ( ={,{a},{b, c},y}. Let f : (X,()((Y,() be the identity map. Then the map f is irresolute but it is not irresolute. Since G ={a,b} is -closed in (Y, (), where f -1 (G)={a, b} is not -closed in X. - Example 4.9: Let X=Y={a, b, c} with topologies ( ={,{a, b},x} and ( = {, {a},{b},{a,b}, Y}. Then the identity map f : (X,()((Y,() is -irresolute but it is not irresolute. Since G = {a} is semi-open in Y, where f -1 (G) ={a} is not semi-open in X. References [1] M E Abd El- Monsef S N El - Deeb and R A Mahmoud, Beta- continuous mapping, Bull. Fac.Sci.Assiut Univ.A 12 (1)(1983), [2] D.Andrijevic, Semi-preopen sets, Mat. Vesnik, 38 (1) (1986), [3] Andrijevic. D, On b-open sets, Mat. Vesink, 48(1996), [4] Balachandran. K. Sundaram. P.and Maki. H. On generalized continuous maps in topological spaces, Mem. Fac.Sci. Kochi Univ. Math. 12 (1991), [5] F.Cammaroto and T.Noiri, Almost irresolute functions, Indian J.Pure. appl. Math, 20(1989), [6] S.G. Crossley and S.K.Hildebrand, semi-topological properties Fund Math; 74(1972), [7] G.Di Maio and T.Noiri, Weak and strong forms of irresolute functions. Rend. Circ. Mat. Palermo(2). Suppl. 18(1988), [8] Ekici. E and Caldas.M, Slightly γ-continuous functions, Bol. Soc. Parana. Mat, (3) 22(2004), 2, [9] G.L.Faro, On strongly α -irresolute mappings, Indian J.Pure appl. Math, 18(1987), [10] K.Kannan and N.Nagaveni, On βˆ -generalized closed sets and open sets in topological spaces, Int. Journal of Math. Analysis, Vol. 6, 57(2012), [11] N.Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, 70 (1963), [12] Levine. N, Generalized closed sets in topology, Rend.Circ. Math. Palermo, 19(2) (1970), [13] S.N. Maheswari and R.Prasad, On α-irresolute maps, Tamkaang J.Math, 11(1980), [14] Mashhour A S, Hasanain I A and El-Deeb S N, ( continuous and ( - open mappings, Acta Math Hungar, 41(1983), [15] O.Njåstad, On some classes of nearly open sets, Pacific J. Math., 15 (1965), [16] A.A.Omari and M.S.M.Noorani, On Generalized b-closed sets, Bull. Malays. Math. Sci.Soc(2),32(1)(2009), [17] P.Sundaram, H.Maki and K.Balachandran, Semi-Generalized continuous maps and semi-t 1/2-spaces, Bull. Fuk. Univ. Edu. Vol 40, Part III(1991),33-40.