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g
-closed sets in topological spaces
Govindappa Navalagi1and Savita B. Megalamani2
1
Department of Mathematics, KIT Tiptur-572202, Karnataka, India
2
Department of Mathematics, Government First Grade College, Davanagere-577004 Karnataka, India
Abstract
In this paper, we introduce and study a new class of generalized closed sets, namely, g -closed sets in topology. Also, we characterize some basic properties of these sets.
Mathematics subject classification(2010); 54A05, 54C08, 54D10.
Key words: -open sets, -closed sets, g -closed sets, g-closed sets, g -closed sets , g -open sets , g -R0 space, g -R1 space.
1.
Introduction
In 1996 D. Andrijevic[1] defined and studied the concepts of b-open sets in topological spaces. b-open sets are also called as sp-open sets. Later in 1997, A.A. El-Atik[5], has
introduced and studied the concept of -open sets in topology. It is known that b-open sets or sp-open sets are same as -open sets. In 2007, E. Ekici[10] has defined and studied the concept of -normal spaces in topology. Also, author has introduced and studied g -closed sets, g-closed sets, , g -closed function and -g-closed function. In 2009, AynurKeskin et.al.[12] has defined and studied the concept of -R0 and -R1 spaces using -open sets. Recently, in 2013, A.I. El-Maghrabi[6] has introduced a concept of g-closed sets in topology which is same as that of g -closed set defined by E. Ekici in 2007. The purpose of this paper is to introduce a new class of generalized closed sets , namely, g -closed sets in topology, also some separation axioms and some related functions.
2.
Preliminaries
In this paper (X, ) and (Y, ) (or X and Y) we always mean topological spaces on which no separation axioms are assumed. Unless otherwise mentioned.
_____________________________________ *Corresponding author
Savita B. Megalamani 2
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International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com For a subset A of X, Cl(A) and Int(A) represent the closure of A and the interior of A
respectively.
The following definitions and results are useful in the sequel:
Definition 2.1: Let X be a topological space. A subset A is called (i)semiopen[13] if ACl(Int(A)),
(ii)preopen[14] if AInt(Cl(A)),
(iii)b-open[1] or sp-open[3] or -open[5] if ACl(Int(A)) Int(Cl(A)).
The complement of semiopen (resp. peropen, b-open or sp-open or -open) set is called semiclosed[2](resp. preclosed[14], b-closed[1] or sp-closed[3] or -closed[5]).
The family of all semiopen (resp. preopen, b-open or sp-open or -open) sets of a space X is denoted by SO(X)(resp. PO(X), BO(X), SPO(X) or O( X)). And the family of all
semiopen(resp. preopen, b-open or sp-open or -open) sets containing a point x of X will be denoted by SO(X,x)(resp. PO(X,x), BO(X,x) or SPO(X,x) or O(X,x)).
Definition 2.2: Let A be a subset of a space X , then semi-interior [2](resp. pre-interior[15], -interior[5]) of A is the union of al semiopen(resp. preopen, -open) sets contained in A and is denoted by sInt(A) (resp. pInt(A), Int(A)).
Definition 2.3: Let A be a subset of a space X , then the intersection of all semi-closed(resp. pre-closed, -closed) sets containing A is called semiclosure[2] (resp. preclosure[4] , -closure[5]) of A and is denoted by sCl(A) (resp. pCl(A), Cl(A)).
Definition 2.4: A subset A of a space X is said to be g -closed[10] if Cl(A)U whenever AU and U.
The complement of g -closed set is said to be g -open.
Definition 2.5: A subset A of a space X is said to be g -closed[6] if Cl(A)U whenever A U and UO( X).
The complement of g -closed set is said to be g -open.
The definitions of be g -closed set and g –closed set respectively , defined by E. Ekici [10] and El-Maghrabi [6] are the same.
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International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com Definition 2.7: A function f: XY is said to be strongly -open[10] , if the image of each -open set of X is -open set in Y.
Definition 2.8: A function f: XY is said to be strongly -closed[10] , if the image of each
-closed set of X is -closed set in Y.
Definition 2.9: A function f: XY is said to be -continuous [5], if the inverse image of each open set of Y is -open set in in X.
Definition 2.10: A function f: XY is said to be -irresolute [7], if the inverse image of each
-open set of Y is -open set in X.
Definition 2.11: A topological space X is called -R0[8] if its every -open set contains the -closure of each of its singletons.
Definition 2.12 : A topological space X is called -R1[9] if for x and y in X with Cl({x}) Cl ({y}), there exist disjoint -open sets U and V such that Cl({x}) is a subset of U and Cl({y}) is a subset of V.
The following are proved in [1]:
Theorem 2.13: For a subset A of a space X the following are equivalent: (i) A is -open
(ii) A=pInt(A) sInt(A) (iii) ApCl(pInt(A)).
Theorem 2.14: Let A be a subset of a space X. Then (i) Cl(A) = sCl(A)pCl(A)
(ii) Int(A) = sInt(A)pInt(A)
(iii)xCl(A) iff AU for every O(X,x) (iv) Cl(A)=A[Int(Cl(A)) Cl(Int(A))] (v)Int(A) =A[Cl(Int(A)) Int(Cl(A))]
Theorem 2.15: Let A be a subset of a space X. Then (i) Cl(Int(A))=Int( Cl(A))= Int(Cl(Int(A))) (ii) Int(Cl(A))=Cl( Int(A))=Cl(Int(Cl(A))) (iii) Cl(sInt(A))=sCl(sInt(A))
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3.
g
-closed sets
In this section , we introduce the following:
Definition 3.1: A subset A of a space X is called generalized star -closed(in brief, g -closed) set if Cl(A)U whenever AU and U is -open set in X.
Definition 3.2: A subset A of a space X is called generalized star -open(in brief, g -open) set if FInt(A) whenever FA and F is -closed set in X.
The family of all g-open sets in topological space X is denoted by g O(X) and that
of , the family of all , g-closed sets in topological space X is denoted by g F(X). And the family of all g -open sets containing x of X will be denoted by gO(X,x)
In view of definitions 2.1(iii), 2.4 and 2.5, we have the following implications:
Remark 3.3: For any subset of a topological space X, we have (i) closedsemiclosed-closed
(ii) closedpreclosed-closed (iii) -closedg -closedg -closed (iv) -closed g -closed
( v ) g-closed g –closed
Clearly, every g-closed set and g -closed set are independent.
We define the following:
Definition 3.4: Let A be a subset of a space X, then the intersection of all g -closed sets containing A is called the g -closure of A and is denoted by g Cl(A).
Definition 3.5: Let A be a subset of a space X, then the union of all g -open sets contained in A is called the g-interior of A and is denoted by g Int(A)
In view of theorem 2.14 and 2.15 we have the following:
Theorem 3.6: Let A be a subset of a space X. Then (i) g Cl(A) = sCl(A) pCl(A)
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International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com (iii) xg Cl(A) iff AU for every gO(X,x)
(iv) g Cl(A)=A[Int(Cl(A)) Cl(Int(A))] (v) g Int(A) =A[Cl(Int(A)) Int(Cl(A))].
Theorem 3.7: Let A be a subset of a space X. Then (i) g Cl(Int(A))=Int(g Cl(A))= Int(Cl(Int(A))) (ii) g Int(Cl(A))=Cl(gInt(A))=Cl(Int(Cl(A))) (iii) g Cl(sInt(A))=sCl(sInt(A))
(iv) g Int(sCl(A))=sInt(sCl(A)).
We define the following:
Theorem 3.8: Atopological space X is calledg -R0, if its every g -open set contains the g -closure of each of its singletons.
Clearly every -R0 spaces is g -R0 spaces.
We characterize the following:
Theorem 3.9: A topological space X is a g-R0 space if and only if for any x and y in X, g Cl({x})g Cl({y}) implies g Cl({x})g Cl({y})=.
Proof: Obvious.
We define the following:
Definition 3.10: A function f: XY is said to be strongly g-closed , if the image of each g -closed set of X is closed set in Y.
Every -closed functions and strongly g -closed functions are independent.
However we have the following:
Lemma 3.11: If f: X Y be -closed function and g:Y Z be strongly g-closed function , then gf is closed function.
Proof: Let F be any closed subset of X. Then f(F) be -closed set in Y, since f: X Y is -closed function. Again, g:YZ is strongly g -closed function and f(F) is -closed set in Y, but we know that every -closed set is g -closed set. And hence f(F) is g -closed set of Y. Then g(f(F))=gf(F) is closed set in Z. This shows that gf is closed function.
156
International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com Definition 3.12: A function f: X Y is called (g , )- closed if the image of each g
-closed set of X is -closed in Y.
Now we give the following:
Lemma 3.13: Let f: X Y be strongly g -closed function and g:Y Z be -closed function, then gf is (g, )- closed function.
Proof: Obvious.
We define the following:
Definition 3.14: A function f: X Y is said to be g -continuous if the inverse image of each open set of Y is g -open set in X.
Definition 3.15: A function f: X Y is said to be g -irresolute if the inverse image of each g-open set of Y is g-open set in X.
We give the following:
Theorem 3.16: If X is a g -R0 space and f : X Y is g -irresolute and strongly g -closed surjection, then Y is g -R0 space.
Proof: Let V be a g -open set of Y and y be any point of V. Since f is g -irresolute , f-1(V) is g -open set in X. Since X is g -R0 space , for a point x f-1({y}), by definition 3.4, g Cl({x})f-1(V) . But by the strongly g -closednessof f, gCl({y}) =g Cl({f(x)})f(g Cl({x}))V. Therefore, Y is g -R0 space.
Theorem 3.17: Let X be a g -R0 space and f : X Y is a strongly g -closed and g -continuous surjection, then Y is an R0 space.
Proof: The proof is similar to the theorem 3.16 and is thus omitted.
Now we define the following:
Definition 3.18: A topological space X is called g -R1 if for x and y in X with g Cl({x})g Cl({y}), there exist disjoint g-open sets U and V such that g Cl({x}) is a subset of U and gCl({y}) is a subset of V.
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International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com We prove the following:
Lemma 3.19: Every g-R1 space is g -R0.
Proof: Let U be a g -open set such that xU. If yU, since xg Cl({y}), we have gCl({x})g Cl({y}). So, there exists a g -open set V such that
g Cl({y})V and xV, which implies yg Cl({x}). Hence g Cl({x})U. Therefore, X is g -R0.
Now, we define the following:
Definition 3.20: A function f: X Y is called :
(i)slightly g -continuous at a point xX if for each clopen subset V in Y containing f(x), there exists a g -open subset U in X containing x such that f(U)V.
(ii)slightly g -continuous if it has this property at each point of X.
We give the following:
Theorem 3.21: Let X Y be topological spaces. The following statements are equivalent for a function f: X Y :
(i) f is slightly g -continuous
(ii) for every clopen set VY , f-1(V) is a g -open (iii) for everyclopen set VY , f-1(V) is a g -closed (iv) for everyclopen set VY , f-1(V) is a g -clopen.
Proof: (i)(ii): Let V be a clopen subset of Y and let xf-1(V). Since f(x) V,
by (i), there exists a g -open set U in X containing x such that Uf-1(V) . We obtain that f-1(V)= U . Thus f-1(V) is g -open.
(ii)(iii): Let V be a clopen subset of Y. Then , Y\V is clopen. By (ii), f-1(Y\V) = X\f-1(V) is a g-open. Thus, f-1(V) is g -closed.
(iii)(iv): Obvious.
(iv)(i): Let V be a clopen subset of Y containing f(x). By (iv), f-1(V) is g -clopen. Take U=f-1(V). Then f(U)V. Hence, f is slightly g-continuous .
We define the following:
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International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com Definition 3.23: A function f : X Y is said to be ( ,g )- continuous if the inverse
image of each -open set of Y is g -open in X.
We recall the following:
Definition 3.24: A function f : X Y is called slightly continuous [11] if the inverse image of each clopen set of Y is open set in X.
Now, We have the following:
Theorem 3.25: Let f : X Y and g:Y Z be functions. Then the following statements are valid:
(i) If f is g -irresolute and g is slightly g -continuous , then gf is slightly g -continuous.
(ii)If f is g-irresolute and g is g -continuous , then gf is g -continuous. (iii)If f is g-irresolute and g is ( , g )-continuous , then gf is ( , g )-continuous.
(iv)If f is g -continuous and g is slightly continuous, then gf is slightly g -continuous.
Proof: (i) Let V be any clopen set in Z. Since g is slightly g -continuous, g-1(V) is g -open in Y. Since, f is g -irresolute and g-1(V) is g -open, then f-1(g-1(V))= ( gf ) -1(V) is g -open in X. Therefore, gf is slightly g -continuous.
(ii) Let V be any open set in Z. Since g is g -continuous, g-1(V) is g -open in Y. Again, f is g -irresolute and g-1(V) is g -open in Y, then f-1(g-1(V))=
( gf ) -1(V) is g -open set in X. Therefore, gf is g -continuous.
(iii) Let V be any -open subset in Z. Since g is ( , g )-continuous, g-1(V) is is g -open in Y. Again, f is g -irresolute and g-1(V) is g -open set in Y, then f-1(g-1(V))=( gf ) -1(V) is g-open set in X. Hence gf is ( , g )-continuous.
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International Journal in Management and Social Sciencehttp://ijmr.net.in, Email: irjmss@gmail.com Theorem 3.26: Let f : X Y and g:Y Z be functions. If f is g -open and surjective and gf : X Z is slightly g -continuous, then g is slightly g -continuous.
Proof: Let V be any clopen set in Z. Since gf is slightly g -continuous, ( gf ) -1(V)= f-1(g-1(V)) is g-open in X. Since, f is g-open, then f(f-1(g-1(V)) )= g-1(V) is g -open in Y. Hence, g is slightly g -continuous.
We define the following:
Definition 3.27: A function f : X Y is said to be always g -open, if the image of each g-open set of X is g -open in Y.
Definition 3.28: A function f : X Y is said to be always g -closed, if the image of each g-closed set of X is g -closed in Y.
Definition 3.29: A function f : X Y is said to be strongly g-open, if the image of each g-open set of X is open in Y.
We have the following:
Theorem 3.30: Let f : X Y and g:Y Z be functions such that gf : X Z is slightly g-continuous
(i)if f is strongly g -open surjective, then g is slightly continuous function. (ii)if f is always g -open surjective, then g is slightly g -continuous.
Proof: (i) Let V be clopen set in Z. Since gf is slightly g -continuous , ( gf ) -1(V)= f-1(g-1(V)) is g-open set in X. Since f is strongly g -open surjective, then
f(f-1(g-1(V)) )=g-1(V) is open set in Y. Hence g is slightly continuous function.
(ii) Let V be clopen set in Z. Since gf is slightly g-continuous, ( gf ) -1(V)= f-1(g-1(V)) is g -open set in X. Since f is always g -open surjective, then
f(f-1(g-1(V)) )=g-1(V) is g -open set in Y. Hence g is slightly g -continuous.
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References
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[7] E. Ekici and Miguel Caldas, “Slightly -continuous functions”, Bol. Soc. Paran. Math., V. 222(2004), 63-74.
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[12] AynurKeskin and TokashiNoiri, “ On -R0 and -R1 spaces”, Miskolc Mathematical notes, vol. 10(2009), no. 2, pp. 137-143.
[13] N. Levine, “Semiopen sets and semi continuity in topological spaces”, Amer. Math. Monthly, 70(1963), 36-41.
