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IJSRR, 7(4) Oct. – Dec., 2018 Page 198

Research article

Available online www.ijsrr.org

ISSN: 2279–0543

International Journal of Scientific Research and Reviews

b**-Compactness And B**-Connectedness In Topological Spaces

Rekha K.

Department of mathematics,

Bishop Heber College, Tiruchirappalli-620017, Tamilnadu, India. Email: rekhamath84@gmail.com

ABSTRACT

In this paper, we introduced and studied new classes of spaces by making use of open cover,

connected and separated sets. We have introduced open cover called b**-open cover in topological

spaces. This paper deals with compact spaces and their properties by using open and

b**-closed sets. The notion of b**-connectedness in topological spaces is also introduced and their

properties are studied. We established relationship between the new spaces and other spaces and also

given examples and characterization.

KEYWORDS:

b**-open sets, b**-closed sets, b**-compact spaces and b**-connectedness.

*Corresponding author

K. Rekha

Department of mathematics,

Bishop Heber College, Tiruchirappalli-620 017, Tamilnadu, India.

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IJSRR, 7(4) Oct. – Dec., 2018 Page 199

INTRODUCTION

The notions of compactness and connectedness are useful and fundamental notions of not only

general topology but also of other advanced branches of mathematics. Many researchers have

investigated the basic properties of compactness and connectedness. The productivity of these

notions of compactness and connectedness motivated mathematicians to generalize these notions.

b-open1 set are introduced by Andrijevic in 1996. The class of b-open sets generates the same topology

as the class of pre-open sets. S. Bharathi, K. Bhuvaneshwari, N. Chandramathi4 introduced

b**-open sets and b**-closed sets in the year 2011. S. S. Benchalli and Priyanka M. Bansali3 introduced

gb-compactness & gb-connectedness in topological spaces in the year 2011. The aim of this paper is

to introduce the concept of b**-compactness & b**-connectedness in topological spaces and is to

give some characterizations of b**-compact spaces.

PRELIMINARIES

Throughout this paper (X,),(Y,)are topological spaces with no separation axioms

assumed unless otherwise stated. LetAX . The closure of Aand the interior ofAwill be denoted

by cl(A)and int(A)respectively.

Definition: 2.1

A subset A of a space X is said to be b-open1 if A Cl(Int(A))Int(Cl(A)).

The complement of b-open set is said to be b-closed. The family of all b-open sets

(respectively b-closed sets) of (X,)is denoted by bO(X,)(respectivelybcl(X,)).

Definition: 2.2

A subset A of a space X is said to be b**-open4 if

))) ( ( ( ))) ( (

(Cl Int A Cl Int Cl A Int

A 

The complement of b**-open set is said to be b**-closed. The family of all b**-open sets

(respectively b**-closed sets) of (X,)is denoted by b**O(X,)

(respectivelyb**cl(X,)).

Definition: 2.3

Let Abe a subset ofX . Then

(i)b-interior of Ais the union of all b-open sets contained in A.

(ii)b-closure of Ais the intersection of all b-closed sets containing A.

(iii)b**-interior of Ais the union of all b**-open sets contained in A.

(iv)b**-closure of Ais the intersection of all b**-closed sets containing A.

Definition: 2.4

A collection {A}Iof subsets of a topological space X is said to be a gb- open

cover3 for X if A X

I

 

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IJSRR, 7(4) Oct. – Dec., 2018 Page 200

Definition: 2.5

A space X is said to be gb- compact if every gb- open cover3 {A}of X

contains a finite sub collection that also coversX .

Definition: 2.6

A topological spacesX is said to be gb-connected3 ifX cannot be expressed as a

disjoint union of two non-empty gb-open sets. A subset of X is gb-connected if it is gb-connected

as a subspace.

Definition: 2.7

A function f :XYis said to be b**-continuous9 if f1(V)is b**-closed in X

for every closed set V ofY.

Definition: 2.8

A function f :XYis said to be b**-irresolute9 if f 1(V)is b**-closed in X

for every *b-closed set V ofY.

* *

b

-COMPACTNESS

Definition: 3.1

A collection {A}Iof subsets of a topological space X is said to be a b**- open

cover for X if A X

I

 

 and Aare b**-open sets in X for eachI.

Definition: 3.2

A space X is said to be b**-compact if every b**-open cover {A}of X contains

a finite sub collection that also coversX .

Theorem: 3.3

Let Ybe a subspace of X .Then Yis b**-compact if and only if every covering of Yby sets

b**-open in X contains a finite sub collection coveringY .

Proof: Let Ybe a subspace ofX . Suppose that Yis b**-compact.

To prove: Every covering of Yby sets b**-open in X contains a finite sub collection

coveringY.

Let V (A)be a covering of Yby sets b**-open inX .

Ais b**-open in X . AYis b**-open in Y.The collection

)} ( * * / {

1 A Y A b O X

V  is covering of Yby sets b**-open inY . Since Yis b**-compact.

There exists a finite sub collection A Y A Y A Y

n

1 , 2 ,..., from V1such that

Y Y A Y A Y A

n  

     ) ( ) ... ( ) ( 2

1  

Y Y A A A

n  

  

( ... )

2

1  

 YA1A2 ...An{A1,A2,...,An}coversY.

} ..., , , { 2

1 A A n

A is a finite sub collection of Vthat coversY . Conversly, Suppose that every

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b**-IJSRR, 7(4) Oct. – Dec., 2018 Page 201

compact. Let {B}be a covering of Y by sets b**-open inY.Bis b**-open in Y . a b**-open

set Ain X such that B  A Y .  B  Afor each  .  B A. Since {B}is a cover

forY. Y B A.  {A}is a cover forYby sets b**-open in X .

By our assumption there exists a finite sub collection

n

A A

A , ,..., 2

1 coveringY.

That is

n

A A

A

Y  1  2 ...  (A1A2 ...An)YY

That is A Y A Y A Y Y

n  

     ) ( ) ... ( ) ( 2

1  

That is B B B Y

n

 

  

1 2 ... , Since BiAiY

 { , ,..., } 2

1 B B n

B   is a finite sub collection of {B}that coversY .

 Every b**-open covering of Y has a finite sub collection coveringY. Therefore Yis

b**-compact.

Theorem: 3.4

Every b**-closed subspace of a b**-compact space is b**-compact.

Proof: Let X be a b**-compact space and Ybe a b**-closed subspace ofX . To prove: Y is

b**-compact.Let {B}be a b**-open cover forY. Bis b**-open in Y  there exists a b**-open set

A in X such that BAY. BAfor all  .  B A {B}is a b**-open cover for

Y.  Y B A. V {A}is a b**-open cover for Ywith b**-open sets in X . Since Yis

b**-closed inX .Therefore, XY is b**-open inX . Consider V1V{(XY)}. V1is a b**-open

cover for X SinceX is b**-compact. There exists a finite sub collection { , ,...} 2 1

2 AA

V  from V1

that coversX .Let V3V2 {(XY)}. That is XYV2 discard it. Now V3covers Y V2is finite

V3 is finite. { , ,..., } 2

1

3 A A A n V

 is a finite sub collection that covers Y .

Y A A A n    

 12 ...   (A1A2 ...An)YY

Y Y A Y A Y A

n  

      ( ) ( ) ... ( ) 2

1  

  B1 B2 ...BnY whereBiAiY. 

} ,..., , {

2

1 B B n

B is a finite sub collection that covers Y.  Every b**-open covering of Yhas a

finite sub collection coveringY. Therefore, Yis b**-compact.

Theorem: 3.5

A b**-continuous image of a b**-compact space is compact.

Proof: Let f :XYbe a b**-continuous map from a b**-compact spaceX onto a topological

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IJSRR, 7(4) Oct. – Dec., 2018 Page 202

Since f is b**-continuous.  f 1(A)is b**-open in X . Then {f 1(A)}Iis a b**-open cover of X . Since X is b**-compact. It has a finite sub cover say ( ), ( 2),..., 1( )

1 1 1

n

A f A f A

f    . Since f is

on-to.  A1,A2,...,Anis a finite cover of Y.  Every open cover of Y has a finite sub cover. Therefore, Yis compact.

Theorem: 3.6

If a map f :XYis b**-irresolute and a subset Bof X is b**-compact relative to X ,then

the image f(B)is b**-compact relative to Y.

Proof: Let {A}Ibe any collection of b**-open subsets ofYsuch that f(B){A,I}

} ),

(

{f 1 A I

B  

 

 . Then B{f1(A),I}. By hypothesis Bis b**-compact

relative toX .Therefore,  a finite subset I0 of IB{f 1(A),I0}. Therefore we have

} ,

{ )

(B A I0

f   . Here {A,I0}is a finite sub collection of {A,I}which covers

) (B f

f(B)is b**-compact relative to Y.

* *

b

-CONNECTEDNESS

Definition: 4.1

Let X be a topological space. A b**-separation of X is a pair A,Bof disjoint

non-empty b**-open subsets of X whose union isX .

That is XAB whereA,B,AB&A,B are b**-open subsets of X .

Note: 4.2

Suppose that A and Bform a b**-separation ofX .

That isXAB, where A,Bare b**-open subsets of X and disjoint.

AandBare both b**-open and b**-closed. Since A and Bare b**-closed.

A A cl

b

 ** ( ) andb**cl(B)B. b**cl(A)BAB. b**cl(B)AAB

Hence Ab**cl(B)andBb**cl(A).

Definition: 4.3

Two subsets Aand Bof a topological space X are said to be b**-separated if

  

B

A , and Ab**cl(B)andBb**cl(A).

Definition: 4.4

A topological space X is said to be b**-disconnected ifXAB,whereA and

Bare any two non-empty b**-separated sets.

Thus, X is b**-disconnected if (i)XAB (ii)(Ab**cl(B))(Bb**cl(A))

(6)

IJSRR, 7(4) Oct. – Dec., 2018 Page 203

Example: 4.6

Let X {a,b,c,d}and  {X,,{c},{d},{c,d},{a,c,d},{b,c,d}}

The collection of b**-open sets of X =

}, , { }, , { }, , { }, , { }, { }, { , ,

{Xc d a c a d b c b d {c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d}}

Let A{a,c}&B{b,d}be non-empty b**-open subsets of X . AB & XAB

A&Bform a b**-separation of X

Example: 4.7

The collection of b**-closed sets of X =

{X,,{a,b,d},{a,b,c},{b,d},{b,c}{a,d},{a,c},{a,b},{d},{c},{b},{a}} Since A{a,c}

b**cl(A){a,c}. B{b,d}

b**cl(B){b,d}

 b**cl(B)

A andBb**cl(A).

A&Bform a b**-separation of X

Example: 4.8

Let X {a,b,c}and {X,,{a}}

The collection of b**-open sets of X = {X,,{a},{a,b},{a,c}}

Let A{a,b}&B{a,c}are non-empty b**-open subsets of X . AB{a} & XAB

The space X is not b**-separated

X is b**-connected.

Theorem: 4.9

Let Cbe a b**-connected subset of a topological spaceX .

LetXAB , where Aand Bare b**-separated subsets of X .Then either CAor CB.

Proof: Let Cbe a b**-connected subset of X andXAB. CXABCAB

)

(A B

C

C   . C(CA)(CB)………….(1)

Since Aand Bare b**-separated sets. Hence Ab**cl(B)andBb**cl(A)

Since Cis b**-connected. Hence, CA or CB[From (1)]

Therefore, CBor CA

Theorem: 4.10

Let

{

C

:

}

be a family of b**-connected subsets of a topological space X such that

C  .Then CCis b**-connected.

Proof: Let

{

C

:

}

be a family of b**-connected subsets of a topological space X such that

(7)

IJSRR, 7(4) Oct. – Dec., 2018 Page 204

LetCAB, where Aand Bare b**-separated subsets of X . Each

C

is b**-connected and

B

A

C

for each . By theorem 4.9, Either

C

A

or

C

B

for each 

Therefore CA

 

 or C Bfor each  . That is CAor CB………(1)

Since

C  . Let xC. Then

x

C

 for each and So xC. Hence xA (or) xB.

But Aand Bare disjoint. Therefore

x

cannot belong to both Aand B

LetxA,xB. Then, by (1), CB. Thus we would haveCA.This contradicts the fact that C

is

b**-disconnected. So Cmust be b**-connected.

Theorem: 4.11

Let X be a space such that each pair of points in X is contained in a b**-connected subset of

X . Then X is b**-connected.

Proof: Let

x

be a given point of X and yx be an arbitrary point inX . By hypothesis, there exists

a b**-connected subset

C

ycontaining

x

andy . Also

X

{

C

y

:

y

X

}

,

{

C

y

:

y

X

}

.

Since each

C

yis b**-connected. By theorem 4.10, X is b**-connected.

Theorem: 4.12

Let Cbe a b**-connected subset of a space X and Let Dbe a subset such that

) ( *

* cl C

b D

C  .Then Dis b**-connected.

Proof: To prove: D is b**-connected. Assume that D is b**-disconnected. Let DAB. Where

Aand Bare b**-separated subsets of X . Then AandB. Since Cis b**-connected and

D

C CAB. By theorem 4.9, Either CA(or) CB. Suppose CA

Then b**cl(C)b**cl(A)b**cl(C)Bb**cl(A)B, Since Aand Bare

b**-separated

ButBDandDb**cl(C) BDBb**cl(C).Hence

  BBDBb**cl(C)

Therefore,  B . B. Which is a contradiction to our assumption.

Hence D must be b**-connected.

Theorem: 4.13

The b**-closure of a b**-connected set is b**-connected.

(8)

IJSRR, 7(4) Oct. – Dec., 2018 Page 205

) ( *

* cl C

b D C 

 . By theorem 4.12, Dis b**-connected. b**cl(C)is b**-connected.

Hence b**-closure of a b**-connected set is b**-connected.

Theorem: 4.14

If Cis a b**-dense subset of a space X and if Cis also b**-connected, then X is

b**-connected.

Proof: Let Cbe b**-dense inX . b**cl(C) XCb**cl(C)CXb**cl(C)

By theorem 4.12, X is b**-connected.

Theorem: 4.15

If f :XYis an on-to b**-continuous map and X is b**-connected, thenY is connected.

Proof: Let f : XYbe an on-to b**-continuous map and X be b**-connected. To prove: Y is

connected. Suppose Y is not connected. Let YABwhere Aand B are disjoint non-empty open

sets inY .

f

1

(

Y

)

f

1

(

A

B

)

f

1

(

Y

)

f

1

(

A

)

f

1

(

B

)

X

f

1

(

A

)

f

1

(

B

)

Since f is b**-continuous andA, B are open sets in Y .

f

1

(

A

)

and

f

1

(

B

)

are b**-open sets in

X .

f

1

(

A

)

f

1

(

B

)

f

1

(

A

B

)

f

1

(

)

X

f

1

(

A

)

f

1

(

B

)

where

f

1

(

A

)

and

)

(

1

B

f

 are disjoint non-empty b**-open sets in X .  Xis not b**-connected. Which is a

contradiction to our assumption. Hence Y is connected.

Theorem: 4.16

If f :XYis an on-to b**-irresolute map and X is b**-connected, thenY is

b**-connected.

Proof: Let f :XYbe an on-to b**-irresolute map and X be b**-connected. To prove: Y is

b**-connected. Suppose Y is not b**-connected. Let YABwhere Aand B are disjoint non-empty

b**-open sets inY .

f

1

(

Y

)

f

1

(

A

B

)

f

1

(

Y

)

f

1

(

A

)

f

1

(

B

)

)

(

)

(

1

1

B

f

A

f

X

Since f is b**-irresolute and Aand B are b**-open sets inY .

f

1

(

A

)

and

f

1

(

B

)

are b**-open

sets in X

f

1

(

A

)

f

1

(

B

)

f

1

(

A

B

)

f

1

(

)

.

X

f

1

(

A

)

f

1

(

B

)

where

f

1

(

A

)

and

)

(

1

B

f

 are disjoint non-empty b**-open sets inX . Xis not b**-connected.

(9)

IJSRR, 7(4) Oct. – Dec., 2018 Page 206

REFERENCES:

1. Andrijivic D , On b-open sets, Mat. Vesnik. 1996; 48(1-2): 59-64.

2. Andrijivic D, Semi pre open sets, Mat. Vesnik. 1986; 38(1): 24-32.

3. Benchali SS and Priyanka M. Bansali, gb-compactness and gb-connectedness, Int. J.

Contemp. Math. Sciences. 2011; 6(10): 465-475.

4. Bharathi S, Bhuvaneshwari K, Chandramathi N, On locally b**-closed sets, International

Journal of Mathematical Sciences and Applications. 2011; 1(2): 636-641.

5. Bourbaki N, General Topology, Part 1, Addison-Wesley (Reading, Mass,1966).

6. Caldas M and Jafari S, On some applications of b-open sets in topological spaces, Kochi, J.

Math. 2007; 2: 11-19.

7. Indira T. and Rekha K, On locally **b-closed sets, Proceedings of the Heber International

Conference on Applications of Mathematics & Statistics (HICAMS), 2012

8. Indira T and Rekha K, *b-t-sets in topological spaces, Antarctica Journal of Mathematics.

2012; 9(7): 599-605.

9. Indira T and Rekha K, On Almost *b-continuous functions, Antarctica Journal of

Mathematics. 2013; 10(1):83-89.

10.Indira T and Rekha K, Decomposition of continuity via *b-open set, Acta Ciencia Indica.

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References

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