eu ON SEMI GENERALIZED STAR b-CLOSED SET IN TOPOLOGICAL SPACES

In this paper, we introduce a new class of sets called semi generalized star b-closed sets in topological spaces (briefly sg∗b-closed set). Also we discuss some of their properties and investigate the relations between the associated topology. AMS Subject Classification: 54A05


Introduction
In 1970, Levine introduced the concept of generalized closed set and discussed the properties of sets, closed and open maps, compactness, normal and separation axioms.Later in 1996 Andrjivic gave a new type of generalized closed set in topological space called b closed sets.The investigation on generalization of closed set has lead to significant contribution to the theory of separation axiom, generalization of continuity and covering properties.A.A. Omari and M.S.M. Noorani made an analytical study and gave the concepts of generalized b closed sets in topological spaces.
In this paper, a new class of closed set called semi generalized star b-closed set is introduced to prove that the class forms a topology.The notion of semi generalized star b-closed set and its different characterizations are given in this paper.Throughout this paper (X, τ ) and (Y, σ) represent the non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned.
Let A ⊆ X, the closure of A and interior of A will be denoted by cl(A) and int(A) respectively, union of all b-open sets X contained in A is called b-interior of A and it is denoted by bint(A), the intersection of all b-closed sets of X containing A is called b-closure of A and it is denoted by bcl(A).
10) a semi generalized closed set (briefly sg-closed) [6] if scl(A) ⊆ U whenever A ⊆ U and U is semi open in X.
11) a generalized pre regular closed set (briefly gpr-closed) [10] if pcl(A) ⊆ U whenever A ⊆ U and U is regular open in X.
12) a semi generalized b-closed set (briefly sgb-closed) [11] if bcl(A) ⊆ U whenever A ⊆ U and U is semi open in X.
13) a g-closed set [19] if cl(A) ⊆ U whenever A ⊆ U and U is sg open in X.

Semi Generalized b Star-Closed Sets
In this section, we introduce semi generalized star b -closed set and investigate some of its properties.Proof.Let A be any g -closed set in X and U be any sg open set containing The converse of above theorem need not be true as seen from the following example.Proof.Let A be any α -closed set in X and U be any sg open set containing The converse of above theorem need not be true as seen from the following example.Proof.Let F be a sg closed set in X such that Definition 5.4.Let x be a point in a topological space X and let x ∈ X.A subset N of X is said to be a sg * b-neighbourhood of x iff there exists a Proof.Let N be a neighbourhood of point x ∈ X.To prove that N is a sg * b-neighbourhood of x.By Definition of neighbourhood, there exists an open set G such that x ∈ G ⊂ N .Hence N is a sg * b-neighbourhood of x.
Remark 5.7.In general, a sg * b-neighbourhood of x ∈ X need not be a neighbourhood of x in X as seen from the following example.Definition 5.12.Let x be a point in a topological space X.The set of all sg * b-neighbourhood of x is called the sg * b-neighbourhood system at x, and is denoted by sg * b-N (x).Theorem 5.13.Let a sg * b-neighbourhood N of X be a topological space and each x ∈ X, Let sg * b-N (X, τ ) be the collection of all sg * b-neighbourhood of x.Then we have the following results.

Theorem 3 . 2 .Theorem 3 . 4 .
Every closed set is sg * b -closed.Proof.Let A be any closed set in X such that A ⊂ U , where U is sg open.Since bcl(A) ⊂ cl(A) = A. Therefore bcl(A) ⊂ U .Hence A is sg * b -closed set in X.The converse of above theorem need not be true as seen from the following example.Example 3.3.Let X = {a, b, c} with τ = {X, φ, {b}, {a, b}}.The set {a, b} is sg * b -closed set but not a closed set.Every g -closed set is sg * b -closed set.

Example 3 . 5 .Example 3 . 7 .Theorem 3 . 8 .
Let X = {a, b, c} with τ = {X, φ, {a, b}}.The set {a, c} is sg * b -closed set but not a g -closed set.Theorem 3.6.Every semi closed set is sg * b -closed set.Proof.Let A be any semi closed set in X and U be any sg open set containing A. Since A is semi closed set, bcl(A) ⊂ scl(A) ⊂ U .Therefore bcl(A) ⊂ U .Hence A is sg * b closed set.The converse of above theorem need not be true as seen from the following example.Let X = {a, b, c} with τ = {X, φ, {a, b}}.The set {b, c} is sg * b -closed set but not a semi closed set.Every α -closed set is sg * b -closed set.

Example 3 . 9 .
Let X = {a, b, c} with τ = {X, φ, {b}, {a, b}}.The set {a, b} is sg * b -closed set but not a α -closed set.Theorem 3.10.Every pre -closed set is sg * b-closed set.Proof.Let A be any pre -closed set in X and U be any sg open set containing A. Since every A pre close set, bcl(A) ⊂ pcl(A) ⊂ U .Therefore bcl(A) ⊂ U .Hence A is sg * b-closed set.The converse of above theorem need not be true as seen from the following example.Example 3.11.Let X = {a, b, c} with τ = {X, φ, {a}, {b}, {a, b}}.The set {a} is sg * b-closed set but not a pre -closed set.Theorem 3.12.Every αg-closed set is sg * b-closed set.Proof.Let A be αg -closed set in X and U be any open set containing A. Since every open set is sg-open sets, we have bcl(A) ⊂ αcl(A) ⊂ U .Therefore bcl(A) ⊂ U .Hence A is sg * b-closed set.The converse of above theorem need not be true as seen from the following example.Example 3.13.Let X = {a, b, c} with τ = {X, φ, {a, b}}.The set {b} is sg * b-closed set but not a αg-closed set.Theorem 3.14.Every sg * b-closed set is gsp-closed set.Proof.Let A be any sg * b-closed set such that U be any open set containing A. Since every open set is sg-open, we have bcl(A) ⊂ spcl(A) ⊂ U .Therefore bcl(A) ⊂ U .Hence A is gsp-closed set.The converse of above theorem need not be true as seen from the following example.Example 3.15.Let X = {a, b, c} with τ = {X, φ, {a}, {a, c}}.The set {a, b} is gsp-closed set but not a sg * b-closed set.Theorem 3.16.Every sg * b-closed set is gb-closed set.Proof.Let A be any sg * b-closed set in X such that U be any open set containing A. Since every open set is sg open, we have bcl(A).Hence A is gb-closed set.The converse of above theorem need not be true as seen from the following example.Example 3.17.Let X = {a, b, c} with τ = {X, φ, {a}, {a, b}}.The set {a, c} is gb-closed set but not a sg * b-closed set.Theorem 3.18.Every sg-closed set is sg * b-closed set.Proof.Let A be any sg-closed set in X such that U be any semi open set containing A. Since every semi open set is sg open, we have bcl(A) ⊂ scl(A) ⊂ U .Therefore bcl(A) ⊂ U .Hence A is sg * b-closed set.The converse of above theorem need not be true as seen from the following example.Example 3.19.Let X = {a, b, c} with τ = {X, φ, {b}, {a, b}}.The set {a, b} is sg * b-closed set but not a sg-closed set.4.Characteristics of sg * b-Closed Sets Theorem 4.1.If a set A is sg * b-closed set then bcl(A) − A contains no non empty sg closed set.
Proof.Since B ⊆ bcl(A), we have bcl(B) ⊆ bcl(A) then bcl(B) − B ⊆ bcl(A) − A. By Theorem 4.1, bcl(A) − A contains no non empty sg closed set.Hence bcl(B) − B contains no non empty sg closed set.Therefore B is sg * b-closed set in X.

Theorem 4 . 4 .
If A is both sg open and sg * b-closed set in X, then A is b closed set.Proof.Since A is sg open and sg * b closed in X, bcl(A) ⊆ A. But A ⊆ bcl(A).Therefore A = bcl(A).Hence A is b closed set.

Theorem 4 . 5 .
For xinX, then the set X − {x} is a sg * b-closed set or sgopen.Proof.Suppose that X − {x} is not sg open, then X is the only sg open set containing X − {x}.(i.e.) bcl(X − {x}) ⊆ X.Then X − {x} is sg * b-closed in X.Note 4.6.g * -closed set and sg * b-closed set are independent to each other as seen from the following examples Example 4.7.Let X = {a, b, c} with τ = {X, φ, {b}, {c}, {b, c}}.The set {b} is sg * b-closed set but not a g * -closed set.

Example 4 . 8 .
Let X = {a, b, c} with τ = {X, φ, {c}, {b, c}}.The set {a, c} is g * -closed set but not a sg * b-closed set.Note 4.9.gp-closed set and sg * b closed set are independent to each other as seen from the following examples.Example 4.10.Let X = {a, b, c} with τ = {X, φ, {c}, {a, c}}.The set {b, c} is gp -closed set but not a sg * b-closed set.

Example 4 . 11 .
Let X = {a, b, c} with τ = {X, φ, {b}, {a, b}}.The set {a, b} is sg * b-closed set but not a gp-closed set.Note 4.12.sg * b closed set and gpr closed set are independent to each other as seen from the following examples.

Example 4 . 13 .
Let X = {a, b, c} with τ = {X, φ, {a}{c}, {a, c}}.The set {a} is sg * b-closed set but not a gpr-closed set.Example 4.14.Let X = {a, b, c} with τ = {X, φ, {c}, {a, c}}.The set {b, c} is spr-closed set but not a sg * b-closed set. 5. Semi Generalized Star b-Open Sets and Semi Generalized Star b-Neighbourhoods In this section, we introduce semi generalized star b-open sets (briefly sg * bopen) and semi generalized star b-neighbourhoods (briefly sg * b-neighbourhood) in topological spaces by using the notions of sg * b-open sets and study some of their properties.Definition 5.1.A subset A of a topological space (X, τ ), is called semi generalized star b-open set (briefly sg * b-open set) if A c is sg * b-closed in X.We denote the family of all sg * b-open sets in X by sg * b-O(X).Theorem 5.2.If A and B are sg * b-open sets in a space X.Then A ∩ B is also sg * b-open set in X. Proof.If A and B are sg * b-open sets in a space X.Then A c and B c are sg * b-closed sets in a space X.By Theorem 4.6

Example 5. 8 .Remark 5 . 9 .
Let X = {a, b, c} with topology τ = {X, φ, {c}, {a, c}}.Then sg * b-O(X) = {X, φ, {c}, {a, c}, {b, c}}.The set {b, c} is sg * bneighbourhood of point c, since the sg * b-open sets {c} is such that c ∈ {c} ⊂ {b, c}.However, the set {b, c} is not a neighbourhood of the point c, since no open set G exists such that c ∈ G ⊂ {b, c}.The sg * b-neighbourhood N of x ∈ X need not be a sg * bopen in X. Theorem 5.10.If a subset N of a space X is sg * b-open, then N is sg * b-neighbourhood of each of its points.Proof.Suppose N is sg * b-open.Let x ∈ N .We claim that N is sg * bneighbourhood of x.For N is a sg * b-open set such that x ∈ N ⊂ N .Since x is an arbitrary point of N , it follows that N is a sg * b-neighbourhood of each of its points.Theorem 5.11.Let X be a topological space.If F is sg * b-closed subset of X and x ∈ F c .Prove that there exists a sg * b-neighbourhood N of x such that N ∩ F = φ.Proof.Let F be sg * b-closed subset of X and x ∈ F c .Then F c is sg * b-open set of X.So by Theorem 5.10 F c contains a sg * b-neighbourhood of each of its points.Hence there exists a sg * b-neighbourhood N of x such that N ⊂ F c .(i.e.) N ∩ F = φ.
finite intersection of sg * b open set is sg * b open.(v) N ∈ sg * b − N (x) ⇒ there exists M ∈ sg * b − N (x) such that M ⊂ N and M ∈ sg * b − N (y) for every y ∈ M. Proof. 1.Since X is sg * b-open set, it is a sg * b-neighbourhood of every x ∈ X.Hence there exists at least one sg * b-neighbourhood (namely-X) for each x ∈ X. Therefore sg * b − N (x) = φ for every x ∈ X.

2 . 1 )Since G 1 ∩
If N ∈ sg * b − N (x), then N is sg * b-neighbourhood of x.By Definition of sg * b-neighbourhood, x ∈ N. 3. Let N ∈ sg * b − N (x) and M ⊃ N .Then there is a sg * b-open set G such that x ∈ G ⊂ N. Since N ⊂ M, x ∈ G ⊂ M and so M is sg * bneighbourhood of x.Hence M ∈ sg * b − N (x).4. Let N ∈ sg * b − N (x), M ∈ sg * b − N (x).Then by Definition of sg * bneighbourhood, there exists sg * b-open sets G 1 and G 2 such that x ∈ G 1 ⊂ N and x ∈ G 2 ⊂ M .Hence x ∈ G 1 ∩ G 2 ⊂ N ∩ M (G 2 is a sg * b-open set, it follows from (1) that N ∩ M is a sg * bneighbourhood of x.Hence N ∩ M ∈ sg * b − N (x).5.Let N ∈ sg * b − N (x), Then there is a sg * b-open set M such that x ∈ M ⊂ N .Since M is sg * b-open set, it is sg * b-neighbourhood of each of its points.Therefore M ∈ sg * b − N (y) for every y ∈ M .