Quasi B-Normal Spaces

In this paper, we introduced the concept of quasi b-normal spaces in topological spaces by using b-open sets and obtained several properties of such space. Andrijevic [1] introduced the concept of b -closed sets and discuss some of their basic properties. Zaitsev [10] introduced the concept of quasi - normal space in topological spaces and obtained several properties of such a space. Recently, Sadeq Ali Saad et al. [7] introduced the concept of quasi p-normal spaces by using p-open sets and obtained several properties of such a space. 2000 AMS Subject Classification: 54D15, 54C08. Key words and phrases: gb-closed, gb-closed sets, b- closed, gb -closed, gb-closed, almost b-closed, almost gb-closed, almost gb-closed,gb-continuous ,almost gb-continuous functions and quasi b- normal spaces

1. Quasi b-Normal Spaces 1.1. Definition. A subset A of a topological space X is said to be 1. b-closed [1] if int(cl(A))  cl(int(A))  A . 2. gb-closed [2] (resp. g*b-closed [9]) if b-cl (A)  U whenever A  U and U is open (resp. g-open) in X. 3. g-closed [4] (resp. gb-closed [3], gp-closed [6], gsp-closed[3] ) if cl(A)  U (resp. b-cl(A)  U, p-cl(A)  U, sp-cl(A)  U )whenever A  U and U is -open in X. The complement of b-closed ( resp. gb-closed, g*b- closed, g - closed, gb-closed, gp-closed, gsp-closed ) set is called b-open ( resp. gb-open, g*b- open, g-closed, gb-open, gp-open,gsp-open) set.The intersection of all b-closed sets containing A is called the b- closure of A and denoted by b-cl(A).The union of all b-open subsets of X which are contained in A is called the b-interior of A and denoted by b-int(A). The family of all gb-closed (resp. gb-open) subsets of the space X is denoted by GBC(X) (resp. GBO(X)). 1.2. Remark [3]. Every b-closed set is gb-closed. 1.3. Proposition [3]. Every gp-closed set is gb-closed. 1.4. Proposition [3]. Every gb-closed set is gsp-closed. 1.5. Remark .We have the following implications for the properties of subsets closed  b-closed g-closed  gp-closed

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gb*-closed  gb-closed  gb-closed  gsp-closed where none of the implications is reversible. 1.6. Example. Let X = {a, b, c } and  = { , X, {a}}. Let A = {c} is a b- closed set but not a closed in X.

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containing A. Hence A is gb-closed, but A is not b- closed, since b-cl (A) = X. 1.8. Example. Let X= {a, b, c}, τ = {X, , {a}, {b}, {a, b}}. Let A= {a}. Then A is b-closed. Hence A is gb-closed, but A is not gp-closed, since A is regular open (- open) and p-cl(A)= {a, c}  A. 1.9. Definition. A space topological X is said to 1. b-normal if for any two disjoint closed subsets A and B of X, there exist two b-open disjoint subsets U and V of X such that A  U and B  V. 2. quasi-normal [10] (resp.quasi b-normal ) if for any two disjoint π-subsets A and B of X, there exist two open(resp. b-open) disjoint subsets U and V of X such that A  U and B  V. 1.10. Remark. For a topological space, the following implications hold: normal  quasi-normal

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b-normal  quasi b-normal. 1.11. Example. Let X = {a, b, c, d} and  = {, {a}, {b}, {a, b}, {c, d}, {a, c, d}, {b, c, d}, X}. The pair of disjoint closed subsets of X are A = {a} and B = {b}. Also U = {a, c} and V = {b, d} are b-open sets such that A  U and B  V. Hence X is b-normal. 1.12. Example. Let X = {a, b, c, d} and  = {,{c}, {d},{c, d},{a, b}, {a, b, c},{a, b, d}, X }. The pair of disjoint - closed subsets of X are A =  and B = {d}. Also U = {c} and V = {a, b, d} are open sets such that A  U and B  V. Hence X is quasi-normal as well as quasi b-normal because every open set is b-open set. 1.13. Corollary [3]. A subset A of a topological space X is gb-open iff F  b-int (A) whenever F  A and F is -closed. 1.14. Theorem. For a topological space X, the following are equivalent: (a) X is quasi b- normal. (b) For any disjoint -closed sets H and K, there exist disjoint gb-open sets U and V such that H  U and K  V. and K  V. (d) For any -closed set H and any -open set V containing H, there exists a gb-open set U of X such that H  U  b-cl(U)  V. (e) For any -closed set H and any -open set V containing H, there exists a gb-open set U of X such that H  U  b-cl(U)  V. Proof. It is obvious that (a)  (b), (b)  (c) and (d)  (e). (c)  (d). Let H be any -closed set of X and V be any -open set containing H. There exist disjoint gb-open sets U, W such that H  U and X  V  W. By Corollary 1.13 , we have X  V  b-int(W) and U  b- int(W) = .Therefore, we obtain b-cl(U)  b-int(W) =  and hence H  U  b-cl(U)  X  b-int(W)  V. (e)  (a). Let H, K be any two disjoint -closed set of X. Then H  X  K and X  K is -open and hence there exists a gb-open set G of X such that H  G  b-cl(G)  X  K. Put U = b-int(G), V = X  b-cl(G). Then U and V are disjoint b-open sets of X such that H  U and K  V. Therefore, X is quasi b- normal. 1.15. Definition. A topological space X is said to be semi-irreducible [5] if every disjoint family of non - empty open sets of X is finite or equivalently if X has only a finite amount of regularly open sets. 1.16. Theorem. A semi-irreducible, semi-regular space X is normal iff X is quasi b- normal. Proof. Let X be quasi b-normal and let A and B be disjoint closed sets of X. By semi-regularity of X, A and B are -closed. Since X is semi-irreducible, X has only finitely many regularly open sets. Thus, A and B are -closed. Now, by the quasi b-normality X is normal. 1.17. Definition. A function f: X → Y is said to be 1. g-closed [4]( resp. g-closed [4], b-closed, gb- closed, gb-closed) if f(F) is g-closed,g-closed, b- closed, gb-closed, gb-closed in Y for every closed set F of X. 2. almost b-closed(resp. almost gb-closed , almost gb-closed ) if f (F) is b-closed (resp. gb-closed, gb- closed ) in Y for every F  RC(X).

of Y. 4. almost continuous [8](resp. almost -continuous [4], almost gb-continuous) if f -1(F) is closed (resp. - closed,gb-closed) in X for every regularly closed set F of Y. From the definitions stated above, we obtain the following diagram: closed  b-closed  gb-closed  gb-closed

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al. closed  al. b-closed  al. gb-closed  al. gb-closed where al. = almost. 1.18. Theorem. A surjection f : X → Y is almost gb-closed if and only if for each subset S of Y and each U  RO(X) containing f -1(S) there exists a gb-open set V of Y such that S  V and f -1(V)  U. Proof. Necessity. Suppose that f is almost gb-closed. Let S be a subset of Y and U  RO(X) containing f – 1(S). If V = Y f (X  U) , then V is a gb-open set of Y such that S  V and f -1(V)  U. Sufficiency. Let F be any regular closed set of X. Then f -1 (Y f (F))  X  F and X – F  RO(X). There exists a gb-open set V of Y such that Y  f (F)  V and f - 1(V)  X  F. Therefore, we have f (F) YV and F  X  f -1(V)  f -1(Y  V) . Hence we obtain f (F) = Y  V and f (F) is gb-closed in Y which shows that f is almost gb-closed. 1.19. Theorem. If f: X → Y is an almost gb-continuous rc-preserving injection and Y is quasi b-normal, then X is quasi b-normal. Proof. Let A and B be any disjoint -closed sets of X. Since f is a rc- preserving injection, f (A) and f (B) are disjoint -closed sets of Y. By the quasi b-normality of Y, there exist disjoint b-open sets U and V of Y such that f (A)  U and f (B)  V. Now if G = int (cl (U)) and H = int (cl (V)), then G and H are disjoint regular open sets such that f (A)  G and f (B)  H. Since f is almost gb-continuous, f -1(G) and f –1(H) are disjoint gb-open sets containing A and B, respectively. It follows from Theorem 1.14 that X is quasi b-normal. Y is b-normal. Proof. Let A and B be any two disjoint closed sets of Y. Then f -1(A) and f -1(B) are disjoint -closed sets of X. Since X is quasi b-normal, there exist disjoint b- open sets of U and V such that f -1 (A)  U and f -1(B)  V. Let G = int(cl(U)) and H = int(cl(V)). Then G and H are disjoint regular open sets of X such that f -1(A)  G and f –1(B)  H. Set K = Y  f (X  G), L = Y f (X  H).Then K and L are b-open sets of Y such that A  K, B  L, f –1(K)  G , f –1(L)  H. Since G and H are disjoint, K and L are disjoint. Since K and L are b- open and we obtain A  b-int(K), B  b-int(L) and b- int(K)  b-int(L) = . Therefore Y is b-normal. 1.21. Theorem. Let f: X →Y be an almost - continuous and almost gb-closed surjection. If X is quasi b-normal space then Y is quasi b-normal. Proof. Let A and B be any disjoint -closed sets of Y. Since f is almost  - continuous, f –1(A), f –1(B) are disjoint closed subsets of X. Since X is quasi b-normal, there exist disjoint b-open sets U and V of X such that f –1(A)  U and f –1(B)  V. Put G = int(cl(U)) and H = int(cl(V)).Then G and H are disjoint regular open sets of X such that f -1(A)  G and f - 1(B)  H. By Theorem 1.18, there exist gb-open sets K and L of Y such that A  K, B  L, f -1 (K)  G and f -1 (L)  H. Since G and H are disjoint. So are K and L by Coollary 1.13, A  b-int(K), B  b-int( L) and b-int(K)  b-int(L) =  . Therefore, Y is quasi b-normal. 1.22. Theorem. Let f: X → Y be an almost continuous and almost gb- closed surjection. If X is normal then Y is quasi b-normal. . Proof. Easy 1.23. Corollary. If f: X → Y is an almost continuous and almost closed surjection and X is a normal space, then Y is quasi b-normal. Proof. Since every almost closed function is almost gb-closed so Y is quasi b-normal.

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