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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
r*bg* -Closed Sets in Topological Spaces.
M.Elakkiya*, N. Sowmya, Asst. Prof N.Balamani.
Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education
for Women,Coimbatore-43,TamilNadu,India.
e-mail:[email protected]* ,[email protected].
ABSTRACT
The aim of this paper is to introduce a new class of sets called r*bg*-closed sets and investigate
some of the basic properties of r*bg*-closed sets.
Index Terms : rbcl, rbint, r*bg*-closed sets, r*bg* -open sets.
I.
INTRODUCTION
Levine [3] introduced the concept of generalised closed sets(briefly g-closed) in topological spaces.
Andrijevic [1] introduced and studied the concepts of b-open sets. Nagaveni and Narmadha[7]
introduced regular b-closed sets. Veerakumar[10] introduced g*- closed sets and studied its properties.
Meenakumari and Indira [5] introduced r*g*-closed sets.The purpose of this paper is to define a new
class of sets called r*bg*- closed sets and also obtain some basic properties of r*bg*-closed sets in
topological spaces.
II. PRELIMINARIES
Definition 2.1 A subset A of a topological space (X,τ) is called
1) A generalized closed (briefly g-closed)[3] set if cl(A)⊆U whenever A⊆U and U is open in (X,τ).
2) a semi -generalized closed (briefly sg-closed)[2] set if scl(A)⊆U and A⊆U and U is semi open in
(X,τ).
3) A generalized α-closed set(briefly gα-closed)[4] if αcl(A)⊆U whenever A⊆U and U is α-open in
(X,τ).
4) b-closed set[1] if cl(int(A))⋂int(cl(A)) ⊆ A .
5) A regular closed set[9] if A=cl(int(A)).
6) A regular generalized closed set(briefly rg-closed)[8] if cl(A)⊂U whenever A⊆U and U is regular
open in (X,τ).
7) a regular b-closed set( briefly rb-closed)[7] if rcl(A)⊂U whenever A⊆ U and U is b-open in(X,τ).
8) an α –generalized regular closed set(briefly αgr- closed)[11] if αcl (A)⊆U whenever A⊆U and U
is regular open in (X,τ).
9) b*-closed set[6] if int(cl(A)) ⊆ U,whenever A⊆U and U is b-open in X.
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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
The complements of the above mentioned closed sets are their respective open sets.
III. r*bg*- CLOSED SETS AND THEIR PROPERTIES
Definition.3.1 A subset A of X is called a r*bg*- closed set if rbcl(A) ⊆ U whenever A ⊆ U, U is b-open.
The class of all r*bg*- closed sets in topological space (X , τ) is denoted by r*bg*-C(X , τ).
Definition.3.2 For a subset A of (X,τ) r*bg* closure of A is denoted by r*bg*cl(A) and is defined as
r*bg*cl(A)=⋂{G;A⊆G, G is r*bg* closed in (X,τ)}.
Example.3.3 Let X={a,b,c} with topology τ={φ,{a},{a,c}, X}. Then r*bg*- closed sets are {φ, {b,c}, X}.
Theorem.3.4. Every regular closed set is r*bg*- closed set.
Proof: Let A be regular closed set in X.Let U be b-open set in X such that A ⊆U.Since A is regular closed
set we have A=cl(int(A)).But cl(int(A)) ⊆ rbcl(A) ⊆ U. Therefore rbcl(A) ⊆ U.Hence A is r*bg* -closed set.
The converse of the above theorem need not be true as seen from the following example.
Example.3.5. Let X={a,b,c} with topology τ={φ, {a}, {b},{a,b},{b,c}, X}. Regular closed sets = {φ,{a},{b, c},
X}. r*bg*- closed sets ={φ,{a},{c},{b, c},{c, a}, X}. The sets {c},{c, a} are r*bg*- closed set but not regular
closed set.
Theorem.3.6. If A is r*bg*-closed set then A is g-closed.
Proof: Let A be r*bg*-closed set in X.Let U open in X such that A ⊆ U.Since A is r*bg*-closed set we have
rbcl(A)⊆ U ,where U is b-open.Now cl(A) ⊆rbcl(A)⊆U.Therefore cl(A) ⊆ U.Hence A is g-closed.
The converse of the above theorem need not be true as seen from the following example.
Example .3.7 Let X={a,b,c} with topology τ={φ, X, {a}, {a,c}};g-closed sets: { φ, X, {b}, {a ,b}, {b, c}};
r*bg*-closed sets: { φ, X, {b,c}}.Then the sets {b} and {a,b} are g-closed set but not r*bg*-closed set.
Theorem .3.8. Every r*bg*-closed set in X is b*-closed set in X.
Proof: Let A be a r*bg*-closed set in X.Let U be b-open set such that A⊆U.Since A is r*bg*-closed set we
have rbcl(A)⊆U. But int(cl(A))⊆ rbcl(A)⊆ U.Therefore int(cl(A)) ⊆ U.Hence A is b*-closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example.3.9 Let X={a,b,c} with topology τ={ φ, X, {a},{b},{a,b}}. r*bg*-closed sets:{ φ, X,{c},{b,c}, {c,a}};
b*-closed sets: { φ, X, {a}, {b}, {c},{b, c}, {c, a}}.Then the sets {a} and {b} are b*-closed set but not r*bg*closed set.
Theorem.3.10. Every r*bg*- closed set is regular generalised (rg) closed set.
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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
Proof: Let A be r*bg*- closed set in X. Let U be regular open set such that A⊆U.Since every regular open
set is b- open and A is r*bg*- closed set, we have cl(A) ⊆ rbcl(A) ⊆ U.Therefore cl(A) ⊆ U.Hence A is rg
closed set in X.
The converse of the above theorem need not be true as seen from the following example.
Example.3.11. Let X={a,b,c} with topology τ={ φ,{a},{b},{a,b}, {b,c}, X }. r*bg*- closed sets:{ φ,{a},
{c},{b,c},{c,a},X}. rg- closed sets:{φ, {a},{b},{c},{a,b},{b,c},{c,a}, X}. Then the sets { b} and {a, b} are rg
closed set but not r*bg*- closed set.
Theorem.3.12 Every r*bg*- closed set is αgr- closed set.
Proof: Let A be r*bg*- closed set in X. Let U be regular open set in X such that A ⊆U.Since every regular
open is b open and A is r*bg*- closed set, we have αcl(A) ⊆ rbcl(A) ⊆ U.Therefore αcl(A) ⊆ U.Hence A is
αgr- closed set.
The converse of the above theorem need not be true as seen from the following example.
Example.3.13. Let X={a,b,c} with topology τ={ φ,{b},{a,c}, X }. r*bg*- closed sets: { φ, {b},{c,a}, X}.αgr closed sets: { φ, {a}, {b},{c},{a,b},{b,c},{c,a}, X}.Then the sets {a}, {c},{a,b},{b,c} are αgr- closed set but not
r*bg*- closed set.
Theorem.3.14. A subset A of X is r*bg*- closed set in X iff rbcl(A)-A contains no non empty b-closed set in
X.
Proof: Let A be r*bg*- closed set.Let rbcl(A)-A contains a b- closed set(say F).Now F ⊆ rbcl(A)-A .Then
F⊆rbcl(A)⋂Ac.Therefore F⊆rbcl(A) and F⊆Ac.. Since Fc is b-open and A is r*bg*-closed set,
rbcl(A)⊆Fc.That is F ⊆ [rbcl(A)]c . Hence F ⊆ rbcl(A)⋂[rbcl(A)]c=φ.F=φ. Thus rbcl(A)-A contains no non
empty b-closed set.
Conversely assume that rbcl(A)-A contains no non empty b-closed set. Let A⊆U,U is b-open.Suppose that
rbcl(A) ⊈ U.Then Uc⋂rbcl(A)≠φ .Then rbcl(A)⋂Uc is a non empty b-closed set and contained in rbcl(A)A.This implies that rbcl(A)-A contains a non empty b-closed set. Which is a contradiction to the fact that
rbcl(A)-A contains no non empty b-closed set.Therefore rbcl(A)⊆U and hence A is r*bg*- closed set.
Theorem.3.15. Suppose that B⊆A⊆X, B is r*bg*-closed set relative to A and that A is both b-open and
r*bg*-closed subset of X, then B is r*bg*-closed set relative to X.
Proof: Let B⊆U and U be an open set in X. It is given that B⊆A⊆X.Therefore B⊆U and B⊆A which implies
that B⊆A⋂ U.Since B is r*bg*-closed set relative to A.rbcl(B)⊆U. That is A⋂rbcl(B)⊆A⋂U, which implies
that A⋂rbcl(B)⊆U.Thus [A⋂rbcl(B)] ⋃[ rbcl(B)]c ⊆ U ⋃[rbcl(B)]c,A⋃[rbcl(B)] c⊆ U ⋃[rbcl(B)]c .Since A is
r*bg*- closed set in X,rbcl(A)⊆U⋃[ rbcl(B)]c . Also B⊆A ⇒ rbcl(B) ⊆ rbcl(A).Thus rbcl(B)⊆ rbcl(A) ⊆ U⋃
[rbcl(B)]c .Therefore rbcl(B) ⊆U and hence B is r*bg*- closed set relative to X.
Theorem.3.16. If A is r*bg*-closed set in X and A⊆B⊆rbcl(A) then B is also r*bg*-closed set in X.
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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
Proof: It is given that A is r*bg*-closed set in X. To prove B is also r*bg*-closed set of X. Let U be an bopen set of X such that B⊆U. Since A⊆B, we have A⊆U.Since A is r*bg*-closed set and rbcl(A)⊆U. Now
rbcl(B)⊆ rbcl( rbcl(A))= rbcl(A)⊆U. So rbcl(B)⊆U. Hence B is r*bg*-closed set.
Theorem.3.17. Let A ⊆ Y⊆ X and suppose that A is r*bg*-closed set in X,then A is r*bg*-closed relative
to Y.
Proof: Let A be r*bg*-closed set in X and let A⊆ Y⋂G,where G is b-open in X.Since A is r*bg*-closed set in
X and A⊆ G, rbcl(A)⊆ G.( ie) Y⋂[ rbcl(A)] ⊆ Y⋂G, which is regular b closure of A in Y.Hence rbclY(A) ⊆
Y⋂G.Therefore A is r*bg*-closed set relative to Y.
Definition.3.18 A subset A of a topological space X is called r*bg*-open set if Ac is r*bg*-closed.
Definition.3.19. For a subset A of (X,τ) r*bg* interior of A is denoted by r*bg*int(A) and is defined as
r*bg*int(A)=⋃{G; G⊆A, G is r*bg*-open in (X,τ)}.
Example.3.20.Consider the topological space X={a,b,c} with the topology τ={φ,{a},{b},{a,b},X }. r*bg*open sets: {φ,{b},{a},{a,b},X}. r*bg* interior sets:{φ,X,{a},{b},{a,b}}.
Theorem.3.21. A set A is r*bg*-open in X iff F ⊆ rbint(A) whenever F is b-closed in X and F⊆ A.
Proof: Assume that A is r*bg*-open in X and F is b-closed set of X such that F⊆ A.Then Ac is r*bg*-closed
set in X and Fc is an b-open set in X containing Ac. Since Ac is r*bg*-closed, rbcl(Ac)⊆ Fc ⇒ [rbint(A)]c ⊆
Fc.Taking complement on both sides, F⊆ rbint(A).
Conversely assume that F ⊆ rbint(A), whenever F ⊆ A and F is b-closed in X.Let G be an open set
containing Ac.Then Gc⊆ rbcl(A).G ⊇ rbint(Ac). rbint(Ac)⊆ G.ie) Ac is r*bg*-closed and hence A is r*bg*open in X.
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International Journal of Advance Foundation and Research in Computer (IJAFRC)
Volume 2, Issue 1, January 2015. ISSN 2348 – 4853
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