Note on contra nano gb-closed maps

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1 Note on contra nano gb-closed maps 1 A. Dhanis Arul Mary and 2 I.Arockiarani Department of Mathematics, Nirmala College for women, Coimbatore, Tamilnadu, India 1 dhanisarulmary@gmail.com, 2 stell11960@yahoo.co.in Abstract In this paper nano gb-closed sets are used to define and investigate a new class of maps, called contra nano gb-closed maps. Relationships between this new class and other class of existing known maps are established AMS Subject Classification: 54A05, 54C08. Keywords: nano gb-closed sets, nano gb-open sets, contra nano gb-closed maps and contra strongly nano gb-closed maps. 1 Introduction Generalized open sets play a very important role in General Topology and they are now the research topics of many topologists worldwide. Indeed a significant theme in general topology concerns the variously modified forms of continuity, separation axioms etc., by utilizing generalized open sets. One of the most well known notions and also an inspiration source is the notion of b-open sets introduced by Andrijevic[1] in Levine [12] introduced the concept of generalized closed sets in topological space. The notion of generalized b-closed sets and its various characterizations were given by Ahmad Al.Omari and Mohd.Salmi Md. Noorani in 2009 [2]. The concept of nano topology was introduced by Lellis Thivagar [13] which was defined in terms of approximations and boundary region of a subset of an universe using an equivalence relation on it. He also established the weak forms of nano open sets namely nano α -open sets, nano semi open sets and nano pre open sets in a nano topological space. Since the advent of these notions several research papers with interesting results in different respects came to existence[4, 5, 6]. In this paper we define a new generalization of maps called contra nano gb-closed maps, contra nano gb-open maps, contra strongly nano gb-closed maps, contra-strongly nano gbopen maps in nano topological space.. Also we study some of their characterizations and JGRMA 2012, All Rights Reserved 55

2 investigate the relationships between such maps. 2 Preliminaries Definition 2.1[19] Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U, R) is said to be the approximation space. Let X U. (i) The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by L R (X). That is, L R (X) = {R(x): R(x) X}, where R(x) denotes the equivalence class determined by X. x U (ii) The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by U R (X). That is, U R (X) = {R(x): R(x) X φ}. x U (iii) The boundary region of X with respect to R is the set of all objects, which can be neither in X nor as not-x with respect to R and it is denoted by B R (X). That is, B R (X) = U R (X) - L R (X). Definition 2.2[19] If (U, R) is an approximation space and X, Y U, then (i) L R (X) X U R (X) (ii) L R (φ) = U R (φ) = φ and L R (U) = U R (U) = U (iii) U R (X Y ) = U R (X) U R (Y ) (iv) U R (X Y ) U R (X) U R (Y ) (v) L R (X Y ) L R (X) L R (Y ) (vi) L R (X Y ) = L R (X) L R (Y ) (vii) L R (X) L R (Y ) and U R (X) U R (Y ) whenever X Y (viii) U R (X c ) = [L R (X)] c and L R (X c ) = [U R (X)] c (ix) U R U R (X) = L R U R (X) = U R (X) (x) L R L R (X) = U R L R (X) = L R (X) Definition 2.3[14] Let U be an universe, R be an aquivalence relation on U and τ R (X) = {U, φ, L R (X), U R (X), B R (X)} where X U. τ R (X) satisfies the following axioms: (i) U and φ τ R (X) (ii) The union of the elements of any sub-collection of τ R (X) is in τ R (X). JGRMA 2012, All Rights Reserved 56

3 (iii) The intersection of the elements of any finite sub-collection of τ R (X) is in τ R (X). That is, τ R (X) forms a topology on U called the nano topology on U with respect to X. We call (U, τ R (X)) as the nano topological space. The elements of τ R (X) are called nano open sets. Definition 2.4[14] Let (U, τ R (X)) be a nano topological space and A U. Then, A is said to be (i) nano semi-open if A Ncl(Nint(A)) (ii) nano pre-open if A Nint(Ncl(A)) (iii) nano α open if A Nint(Ncl(Nint(A))) (iv) nano semi pre-open if A Ncl(Nint(Ncl(A))) (v) Nano b-open if A Ncl(Nint(A)) Nint(Ncl(A)) Definition 2.5 A subset A of a topological space (U, τ R (X)) is called (i) nano generalized closed (briefly, nano g-closed)[4] if Ncl(A) G whenever A G and G is nano open in U. (ii) nano semi-generalized closed (briefly nano sg-closed)[5] if N scl(a) G whenever A G and G is nano semi-open in U. (iii) nano generalized b-closed (briefly nano gb-closed)[18], if Nbcl(A) G whenever A G and G is nano open in U. Definition 2.6[15] Let (U, τ R (X)) and (V, τ R (Y )) be nano topological spacces, then a map f:(u, τ R (X)) (V, τ R (Y )) is said to be (i) nano continuous if f 1 (V ) is nano closed in (U, τ R (X)) for each nano closed set V in (V, τ R (Y )). (ii) nano b-continuous if f 1 (V ) is nano b-closed in (U, τ R (X)) for each nano closed set V in (V, τ R (Y )). 3 Contra nano gb-closed and Contra-strongly nano gb-closed maps This section defines two types of contra nano gb-closed maps through nano gb-closed sets, derives their properties and establishes their relationship. JGRMA 2012, All Rights Reserved 57

4 Definition 3.1 A map f:(u, τ R (X)) (V, τ R (Y )) is said to be contra nano gb-closed map (resp. contra nano gb-open) if the image of every nano closed (resp. nano open) set in (U, τ R (X)) is a nano gb-open (resp. nano gb-closed) set in (V, τ R (Y )). Remark 3.2 Contra nano gb-closedness and nano gb-closedness are independent notions as shown in the following example. Example 4.3 Let U = {a, b, c, d} with U/R = {{a}, {c}, {b, d}} and X = {a, b}, then τ R (X) = {{U, φ,{a}, {a, b, d}, {b, d}}.let V = {x, y, z, w} with V/R = {{x}, {y, w}, {z}} and Y = {y, w}, then τ R (Y ) = {U, φ, {y, w}}. Define f:u V as f(a) = y, f(b) = y, f(c) = w, f(d) = x. Then, f is contra nano gb-closed but not nano gb-closed because {a, c} is nano closed in U, but f({a, c}) = {y, w} is not nano gb-closed in V. Example 3.4 Let U = {a, b, c, d} with U/R = {{a}, {b, d}, {c}} and X = {b, d}, then τ R (X) = {{U, φ, {b, d}}.let V = {x, y, z, w} with V/R = {{x}, {y, w}, {z}} and Y = {x, y}, then τ R (Y ) = {U, φ, {x}, {x, y, w}, {y, w}}. Define f:u V as f(a) = z, f(b) = y, f(c) = z, f(d) = w. Then, f is nano gb-closed but not contra nano gb-closed because {a, c} is nano closed in U, but f({a, c}) = {z} is not nano gb-open in V. Similarly, contra nano gb-openness and nano gb-openness are independent notions. Theorem 3.5 Let (U, τ R (X)) and (V, τ R (Y )) are nano topological spaces. Then if (i) f:(u, τ R (X)) (V, τ R (Y )) is contra nano gb-closed and A is nano closed subset of (U, τ R (X)), then f A :(A, U A ) (V, τ R (Y )) is contra nano gb-closed. (ii) f:(u, τ R (X)) (V, τ R (Y )) is contra nano gb-open and A is nano open subset of (U, τ R (X)), then f A :(A, U A ) (V, τ R (Y )) is contra nano gb-open. (iii) f:(u, τ R (X)) (V, τ R (Y )) is contra nano gb-closed bijective function and A = f 1 (B) for some nano open subset B of (V, τ R (Y )), then f A :(A, U A ) (V, τ R (Y )) is contra nano gb-closed. Proof (i)let B be a nano closed set of A. Then, B = A F for some nano closed set F of (U, τ R (X)) and so B is nano closed in (U, τ R (X)). By hypothesis, f(b) is nano gb-open. But f(b) = f A (B) and therefore, f A is contra nano gb-closed. (ii)let B be a nano open set of A. Then, B = A G for some nano open set G of (U, τ R (X)) and so B is nano open in (U, τ R (X)). By hypothesis, f(b) is nano gb-closed. But f(b) = f A (B) and therefore, f A is contra nano gb-open. (iii)let D be a nano closed set of A. Then, D = A H for some nano closed set H of (U, JGRMA 2012, All Rights Reserved 58

5 τ R (X)). But f(d) = f A (D) = f(a H) = f(f 1 (B) H) = B f(h). Since, f is contra nano gb-closed f(h) is nano gb-open and so B f(h) is nano gb-open in (V, τ R (Y )). Hence f is contra nano gb-closed. Theorem 3.6 Let f:(u, τ R (X)) (V, τ R (Y )) and g:(v, τ R (Y )) (W, τ R (Z)) be any two maps such that g f:(u, τ R (X)) (W, τ R (Z)) is contra nano gb-open map. If f is contra nano continuous and surjective, then g is nano gb-closed. Proof Let A be nano closed set in (V, τ R (Y )). Since f is contra nano continuous f 1 (A) is nano open in (U, τ R (X)). Then, (g f)f 1 (A) is nano gb-closed in (W, τ R (Z)). That is, g(a) is nano gb-closed in (W, τ R (Z)). Therefore, g is nano gb-closed map. Definition 3.7 A map f:(u, τ R (X)) (V, τ R (Y )) is said to be contra strongly nano gb-closed map (resp. contra strongly nano gb-open) if the image of every nano gb-closed (resp. nano gb-open) set in (U, τ R (X)) is a nano gb-open (resp. nano gb-closed) set in (V, τ R (Y )). Theorem 3.8 (i)every contra strongly nano gb-closed map is contra nano gb-closed. (ii) Every contra strongly nano gb-open map is contra nano gb-open. Proof (i) Let f be contra strongly nano gb-closed map and A be a nano closed set in (U, τ R (X)). Since, every nano closed set is nano gb-closed, A is nano gb-closed and f(a) is nano gb-open in (V, τ R (Y )). Hence f is contra nano gb-closed. (ii) Proof is similar to the proof of (i). Remark 3.9 The implications in Theoem 5.8 are not reversible in general as seen from the following examples. Example 3.10 Let U = {a, b, c, d} with U/R = {{a}, {c}, {b, d}} and X = {a, b}, then τ R (X) = {{U, φ,{a}, {a, b, d}, {b, d}}.let V = {x, y, z, w} with V/R = {{x}, {y, w}, {z}} and Y = {y, w}, then τ R (Y ) = {U, φ, {y, w}}. Define f:u V as f(a) = x, f(b) = z, f(c) = y, f(d) = w. Then, f is contra nao gb-closed but not contra-strongly nano gb- closed because {a, b} is nano gb-closed in U, but f({a, b}) = {x, z} is not nano gb-open in V. Example 3.11 Let U = {a, b, c, d} with U/R = {{a}, {b, d}, {c}} and X = {b, d}, then τ R (X) = {{U, φ, {b, d}}. Let V = {x, y, z, w} with V/R = {{x}, {y, w}, {z}} and Y = {x, y}, then τ R (Y ) = {U, φ, {x}, {x, y,w}, {y, w}}. Define f:u V as f(a) = x, f(b) = y, f(c) = w, f(d) = z. Then, f is contra nano gb-open but not contra-strongly nano gb-open because JGRMA 2012, All Rights Reserved 59

6 {a, b, c} is nano gb-open in U, but f({a, b, c}) = {x, y, w} is not nano gb-open in V. Theorem 3.12 If the map is bijective, then contra-strongly nano gb-closedness and contrastrongly nano gb-openness are equivalent. Proof Let f be contra-strongly nano gb-closed and G be a nano gb-open set in (U, τ R (X)), then G c is nano gb-closed and f(g c ) is nano gb-open in (V, τ R (Y )). That is, f(g) is nano gb-closed in (V, τ R (Y )). Hence f is contra-strongly nano gb-open. Similarly the other part can be proved. Theorem 3.13 For a map f:(u, τ R (X)) (V, τ R (Y )) the following are equivalent if the arbitrary union of class of nano gb-open sets in (V, τ R (Y )) is also nano gb-open. (i) f is contra nano gb-open. (ii) For every subset B of (V, τ R (Y )) and for every nano gb-closed subset F of (U, τ R (X)) with f 1 (B) F, there exists a nano gb-open subset G of (V, τ R (Y )) with B G and f 1 (G) F. (iii) For every v V and for every nano gb-closed subset F of U with f 1 (v) F, there exists a nano gb-open subset G of V with v G and f 1 (G) F. Proof (i) (ii)let B be a subset of V and F be a nano gb-closed subset of U with f 1 (B) F. Since f is contra-strongly nano gb-open and F c is nano gb-open subset of U, f(f c ) is nano gb-closed subset of V. Then (f(f c )) c is nano gb-open subset of V. Put G = (f(f c )) c. Then G is a nano gb-open subset of V and since f 1 (B) F we have, f(f c ) B c and hence B G. Moreover f 1 (G) = f 1 (f(f c )) c F (ii) (iii) It is sufficient to put B = {v}. (iii) (i) Let A be a nano gb-open subset of U and v (f(a)) c and let F = A c. Then F is a nano gb-closed subset of U. By (iii), there exists a nano gb-open subset G v of V with v G v and f 1 (G v ) F. Then v G v (f(a)) c. Hence (f(a)) c = {G v /v (f(a)) c } is nano gb-open. Therefore, f(a) is nano gb-closed subset of V. Hence f is contra-strongly nano gb-open. Theorem 3.14 For a map f:(u, τ R (X)) (V, τ R (Y )) the following are equivalent. (i) f is contra-strongly nano gb-closed. (ii) For every subset B of V and for every nano gb-open subset G of U with f 1 (B) G, there exists a nano gb-closed subset F of V with B F and f 1 (F ) G. Proof (i) (ii) Let B be a subset of V and G be a nano gb-open subset of U with f 1 (B) G. Since f is contra-strongly nano gb-closed and G c is nano gb-closed subset of U, f(g c ) is nano gb-open subset of V. Then (f(g c )) c is nano gb-closed subset of V. Put F = (f(g c )) c. Then JGRMA 2012, All Rights Reserved 60

7 F is a nano gb-closed subset of V and since f 1 (B) G we have, f(g c ) B c and hence B F Moreover f 1 (F ) = f 1 ((f(g c )) c ) G. (ii) (i) Let A be a nano gb-closed subset of U. Put B= (f(a)) c and let G = A c. Hence f 1 (B) = f 1 ((f(a)) c ) A c = G. By assumption there exists a nano gb-closed subset F of V with B F and f 1 (F ) G. Then B = F. If v F and v/ B, then v f(a). Therefore, v = f(u) for some u A and we have that u f 1 (F ) G = A c. Which is a contradiction. Since, B = F, (f(a)) c = F. That is, (f(a)) c is nano gb-closed and hence f(a) is a nano gb-open set in V. Hence f is contra-strongly nano gb-closed. Corollary 3.15 If f:(u, τ R (X)) (V, τ R (Y )) is contra-strongly nano gb-closed then for every v V and for every nano gb-open subset G of U with f 1 (v) G, there exists a nano gb-closed subset F of V with v F and f 1 (F ) G. Proof follows by taking the set B in the theorem 5.17 to be {v} for v V. Remark 3.16 Contra-strongly nano gb-closedness and strongly nano gb-closedness are independent notions as seen from the following examples. Example 3.17 Let U = {a, b, c, d} with U/R = {{a, d}, {b}, {c}} and X = {a,c}, then τ R (X) = {{U, φ, {c}, {a, c, d}, {a, d}}.let V = {x, y, z, w} with V/R = {{x}, {y}, {z}, {w}} and Y = {x, w}, then τ R (Y ) = {U, φ, {x, w}}. Define f:u V as f(a) = x, f(b) = x, f(c) = w, f(d) = z. Then, f is contra-strongly nano gb-closed but not strongly nano gb-closed because {a, b, c} is nano gb-closed in U, but f({a, b, c}) = {x, w} is not nano gb-closed in V. Example 3.18 Let U = {a, b, c, d} with U/R = {{a}, {b, c}, {d}} and X = {a,c}, then τ R (X) = {{U, φ, {a}, {a, b, c}, {b, c}}.let V = {x, y, z, w} with V/R = {{x, z}, {y}, {w}} and Y = {x, w}, then τ R (Y ) = {U, φ, {w}, {x, z, w}, {x, z}}. Define f:u V as f(a) = y, f(b) = x, f(c) = y, f(d) = z. Then, f is strongly nano gb-closed but not contra-strongly nano gb-closed because {a} is nano gb-closed in U, but f({a}) = {y} is not nano gb-open in V. Similarly, the concepts of contra-stronglyy nano gb-openness and strongly nano gb-openness are independent. Remark 3.19 The composition of two contra-strongly nano gb-closed maps need not be contra-strongly nano gb-closed as seen from the following example. JGRMA 2012, All Rights Reserved 61

8 Example 3.20 Let U = {x, y, z, w}= W with U/R = {{x}, {y}, {z}, {w}}= W/R and X = Z = {x, w}, then τ R (X) =τ R (Z) = {{U, φ, {x, w}}.let V = {a, b, c, d} with V/R = {{a, d}, {b}, {c}} and Y = {a, c}, then τ R (Y ) = {U, φ, {c}, {a, c, d}, {a, d}}. Define f:u V as f(x) = a, f(y) = c, f(z) = a, f(w) = d and g:v W as g(a) = x, g(b) = y, g(c) = w, g(d) = z. Then, f and g are contra-strongly nano gb-closed maps but g f: (U, τ R (X)) (W, τ R (Z)) is not contra-strongly nano gb-closed because (g f)({x, y}) = {x, w} is not nano gb-open in (W, τ R (Z)). Remark 3.21 The composition of two contra-strongly nano gb-open maps need not be contra-strongly nano gb-open as seen from the following example. Example 3.22 Let U = {a, b, c, d} = W with U/R = {{a}, {b, c}, {d}} = W/R and X =Z= {a,c}, then τ R (X) =τ R (Z) = {{U, φ, {a}, {a, b, c}, {b, c}}.let V = {x, y, z, w} with V/R = {{x, z}, {y}, {w}} and Y = {x, w}, then τ R (Y ) = {U, φ,{w}, {x, z, w}, {x, z}}. Define f:u V as f(a) = y, f(b) = x, f(c) = w, f(d) = x and g:v W as g(x) = c, g(y) = a, g(z) = b, g(w) = d. Then, f and g are contra-strongly nano gb-open maps but g f: (U, τ R (X)) (W, τ R (Z)) is not contra-strongly nano gb-open because (g f)({a, b}) = {a, c} is not nano gb-closed in (W, τ R (Z)). Theorem 3.23 Let f:(u, τ R (X)) (V, τ R (Y )) and g:(v, τ R (Y )) (W, τ R (Z)) be any two maps such that g f:(u, τ R (X)) (W, τ R (Z)) (i) If f is strongly nano gb-closed and g is contra strongly nano gb-closed then g f is contra strongly nano gb-closed. (ii) If f is contra-strongly nano gb-closed and g is strongly nano gb-open then g f is contrastrongly nano gb-closed. Proof (i) Let G be nano gb-closed in (U, τ R (X)). Then f(g) is nano gb-closed in (V, τ R (Y )) and g(f(g)) is nano gb-open in (W, τ R (Z)). Hence g f is contra-strongly nano gb-closed. (ii) Let G be nano gb-closed in (U, τ R (X)). Then f(g) is nano gb-open in (V, τ R (Y )) and g(f(g)) is nano gb-open in (W, τ R (Z)). Hence g f is contra-strongly nano gb-closed. Theorem 3.24 Let f:(u, τ R (X)) (V, τ R (Y )) be a map. (i) If f is contra-strongly nano gb-open, then Ngbcl(f(G)) f(ngbcl((g)) for every nano gbopen subset G of (U, τ R (X)). (ii) If f is contra-strongly nano gb-closed, then f(g) Ngbint(f(Ngbcl((G))) for every subset G of (U, τ R (X)). JGRMA 2012, All Rights Reserved 62

9 Proof (i) If f is contra-strongly nano gb-open, f(g) is nano gb-closed in (V, τ R (Y )) for every nano gb-open set G in (U, τ R (X)). Then, Ngbcl(f(G)) = f(g) f(ngbcl((g)) (ii) Since f is strongly nano gb-closed and Ngbcl(G) is nano gb-closed, f(ngbcl(g)) is nano gb-open. we have, f(g) f(ngbcl(g)) = Ngbint(f(Ngbcl((G))). Theorem 3.25 Let f:(u, τ R (X)) (V, τ R (Y )) and g:(v, τ R (Y )) (W, τ R (Z)) be any two maps such that g f:(u, τ R (X)) (W, τ R (Z)) (i) If f is strongly nano gb-open and g is contra strongly nano gb-open then g f is contra strongly nano gb-open. (ii) If f is contra-strongly nano gb-open and g is strongly nano gb-closed then g f is contrastrongly nano gb-open. Proof (i) Let G be nano gb-open in (U, τ R (X)). Then f(g) is nano gb-open in (V, τ R (Y )) and g(f(g)) is nano gb-closed in (W, τ R (Z)). Hence g f is contra-strongly nano gb-open. (ii) Let G be nano gb-open in (U, τ R (X)). Then f(g) is nano gb-closed in (V, τ R (Y )) and g(f(g)) is nano gb-closed in (W, τ R (Z)). Hence g f is contra-strongly nano gb-open. Theorem 3.26 Let f:(u, τ R (X)) (V, τ R (Y )) and g:(v, τ R (Y )) (W, τ R (Z)) be any two maps such that g f:(u, τ R (X)) (W, τ R (Z)) is contra strongly nano gb-closed (i) If f is a nano gb-irresolute surjection then g is contra strongly nano gb-closed. (ii) If g is nano gb-irresolute injection then f is contra strongly nano gb-closed. Proof (i) Let A be nano gb-closed in (V, τ R (Y )). Since f is nano gb-irresolute f 1 (A) is nano gb-closed in (U, τ R (X)). since, g f is contra-strongly nano gb-closed and f is surjective (g f)(f 1 (A)) = g(a) is nano gb-open in (W, τ R (Z)). Hence g is contra-strongly nano gb-closed. (ii) Let A be nano gb-closed in (U, τ R (X)) Since, g f is contra-strongly nano gb-closed, (g f)(a) is nano gb-open in (W, τ R (Z)). Since g is nano gb-irresolute injection, g 1 ((g f)(a)) = f(a) is nano gb-open in (V, τ R (Y )). Hence f is contra-strongly nano gb-closed. Theorem 3.27 Let f:(u, τ R (X)) (V, τ R (Y )) and g:(v, τ R (Y )) (W, τ R (Z)) be any two maps such that g f:(u, τ R (X)) (W, τ R (Z)) is contra strongly nano gb-open (i) If f is a nano gb-irresolute surjection then g is contra strongly nano gb-open. (ii) If g is nano gb-irresolute injection then f is contra strongly nano gb-open. Proof (i) Let A be nano gb-open in (V, τ R (Y )). Since f is nano gb-irresolute f 1 (A) is nano gb-open in (U, τ R (X)). Since, g f is contra-strongly nano gb-open and f is surjective (g f)(f 1 (A)) = g(a) is nano gb-closed in (W, τ R (Z)). Hence g is contra-strongly nano JGRMA 2012, All Rights Reserved 63

10 gb-open. (ii) Let A be nano gb-open in (U, τ R (X)). Since, g f is contra-strongly nano gb-open, (g f)(a) is nano gb-closed in (W, τ R (Z)). Since g is nano gb-irresolute injection, g 1 (g f)(a)) = f(a) is nano gb-closed in (V, τ R (Y )). Hence f is contra-strongly nano gb-open. Definition 3.28 A map f:(u, τ R (X)) (V, τ R (Y )) is said to be contra nano gb-irresolute if the inverse image of every nano gb-open set in (V, τ R (Y )) is nano gb-closed in (U, τ R (X)). Theorem 3.29 Let f:(u, τ R (X)) (V, τ R (Y )) and g:(v, τ R (Y )) (W, τ R (Z)) be any two maps such that g f:(u, τ R (X)) (W, τ R (Z)) (i) If g f is strongly nano gb-closed map and g is contra nano gb-irresolute injective then f is contra-strongly nano gb-closed. (ii) If g f is contra nano gb-irresolute and g is contra-strongly nano gb-closed injective then f is nano gb-irresolute. Proof (i) Let A be nano gb-closed in (U, τ R (X)). Then (g f)(a) is nano gb-closed in (W, τ R (Z)). Since g is contra nano gb-irresolute, g 1 ((g f)(a)) is nano gb-open in (V, τ R (Y )). that is, f(a) is nano gb-open in (V, τ R (Y )). Hence f is contra-strongly nano gb-closed. (ii) Let A be nano gb-closed in (V, τ R (Y )). Since g is contra-strongly nano gb-closed, g(a) is nano gb-open in (W, τ R (Z)). Since g f is contra nano gb-irresolute, (g f) 1 (g(a)) is nano gb-closed in (U, τ R (X)). That is, f 1 (A) is nano gb-closed in (U, τ R (X)). Hence f is nano gb-irresolute. References [1] D. Andrijevic, On b-open sets, Mat. Vesnik 48(1996), no.1-2, [2] Ahmad Al-Omari and Mohd. Salmi Md. Noorani, On generalized b-closed sets, Bull. Malays. Math. Sci. Soc. (2)32(1)(2009), [3] Ahmed I. El-maghrabi, More on γ-generalized closed sets in topology, Journal of Taibah University for science 7(2013), [4] Bhuvaneswari and Mythili Gnanapriya K, Nano generalized closed sets in nano topological spaces,international Journal of Scientific and Research Publications, (2014). JGRMA 2012, All Rights Reserved 64

11 [5] Bhuvaneswari K, Ezhilarasi, On nano semi- generalized and generalized- semi closed sets in nano topological spaces, International Journal of Mathematics and Computer Applications Research, (2014), [6] M. Caldas and S. Jafari, On some applications of b-open sets in topological spaces, Kochi, J. Math.2(2007), [7] J. Donntchev, Contra continuous functions and strongly-s closed spaces, Internat. J. Math. Sci. 19(1996), [8] A. Dhanis Arul Mary and I. Arockiarani, Properties of nano gb-closed maps, IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 11, Issue 2 Ver. II (Mar - Apr. 2015), PP [9] E. Ekici and Caldas, Slightly b-continuous functions, Bol. Soc. Parana. Mat. (3) 22 (2004), no. 2, [10] M. Ganster and M. Steiner, On b -closed sets, Appl. Gen. Topol. 8 (2007), no. 2, [11] M. Ganster and M. Steiner, On some questions about b-open sets, Questions Answers Gen.Topology 25(2007), no.1, [12] M. Ganster and I. L. Reilly, A Decomposition of continuity, Acta math. Hungar. 56(1990), [13] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo (2) 19 (1970), [14] M. Lellis Thivagar and Carmel Richard, On nano forms of weakly open sets, International Journal of Mathematics and statistics Invention, (2013), [15] M. Lellis Thivagar and Carmel Richard On Nano Continuity, Mathematical theory and modeling, (2013), no.7, [16] T. Noiri, R. Sayed On Decomposition of continuity, Acta math. Hungar. 111(1-2)(2006), 1-8. [17] A. A. Nasef, some properties of contra γ-continuous functions, Chaos Solitons Fractals 24(2005), no.2, JGRMA 2012, All Rights Reserved 65

12 [18] R.Thanga Nachiyar, K.Bhuvaneswari, Nano Generalized α - Continuous and Nano α- Generalized Continuous Functions in Nano Topological Spaces, International Journal of Mathematics Trends and Technology ISSN: , Volume 14Number 2Oct2014. [19] Z. Pawlak (1982) Rough Sets, International Journal of Information and Computer Sciences, 11(1982), JGRMA 2012, All Rights Reserved 66