Semi - α - Compactness in Bitopological Spaces

2008

Semi - - Compactness in Bitopological SpacesBy

Qays H. I. Al-Rubaye

Al-Muthanna University, College of Education, Department of Mathematics

Abstract :In this paper, we study especial case of compactness in bitopological spaces by considering ij - semi - -

open sets , we prove some results about them comparing with similar cases in topological spaces .

Key words : Bitopological space , ij - semi - - open set, ij - semi - - compact space .

:الخالصة شبھ-ijاالعتماد على مجموعاتمن خالل ثنائیة التبولوجيمن التراص في الفضاءاتةدراسة حالة خاصتمفي ھذا البحث

.، أثبتنا بعض النتائج عنھا مقارنة مع حاالت مماثلة في الفضاءات التبولوجیةألفا المفتوحة

1. Introduction :The study of bitopological spaces was initiated by [Kelly, J .C ., 1963] . A triple

),,( 21 X is called bitopological space if ),(and),( 21 XX are two topological spaces . In

1990 ,Jelic , M ., introduced the concept of setsopen--ij in bitopological spaces . In 1981 ,

Bose ,S. ,introduced the notion of setsopen-semi-ij in bitopological spaces . In 1992 , Kar A .

, have introduced the notion of setsopen-pre-ij in bitopological spaces . In 2012 , H . I . Al-

Rubaye , Qaye , introduced the notion of open--semi- ij sets in bitopological spaces .

In this paper , we study especial cases of compactness in bitopological spaces by

considering open--semi- ij sets and their relationships with compact--ij space and

compact-pre-ij space are studied .

2. Preliminaries :Throughout the paper , spaces always mean a bitopological spaces , the closure and the

interior of any subset A of X with respect to i , will be denoted by Acli , and Ai int

respectively, for 2,1i .

Journal of Babylon University/ Pure and Applied Sciences/ No.(6)/ Vol.(21): 2013

2009

Definition 2.1 :

Let ),,( 21 X be a bitopological space , XA , A is said to be :

(i) setopen-pre-ij [Kar A . , 1992] if ))(int( AcljiA , where 2,1,; jiji ,

(ii) setopen-semi-ij [Bose , S . , 1981] if ))int(( AicljA , where 2,1,; jiji ,

(iii) setopen--ij [Kumar Sampath , S. , 1997] if )))int((int( AicljiA , where

2,1,; jiji .

Remark 2.2 :

The family of open-pre-ij ( resp. open-semi-ij and open--ij ) sets of X is denoted by

)(- XPOij ( resp. )(- XSOij and )(- XOij ) , where 2,1,; jiji .

Example 2.3 :

Let },,{ cbaX , }}{,,{1 aX , and }},{},{},{,,{2 cbcbX .

),(and),( 21 XX are two topological spaces , then ),,( 21 X is a bitopological space .

The family of all setsopen-pre-21 of X is : }},{},{,,{)(-12 cbaXXPO .

The family of all setsopen-semi-21 of X is : }}{,,{)(-12 aXXSO .

The family of all setsopen--21 of X is : }}{,,{)(-12 aXXO .

Definition 2.4 :

The complement of an open-pre-ij ( resp. open-semi-ij and open--ij ) set is said to

be closed-pre-ij ( resp. closed-semi-ij and closed--ij ) set . The family of closed-pre-ij

( resp. closed-semi-ij and closed--ij ) sets of X is denoted by )(- XPCij ( resp. )(- XSCij

and )(- XCij ) , where 2,1,; jiji .

Remark 2.5 :

It is clear by definition that in any bitopological space the following hold :

(i) every issetopeni setopen--,open-semi-,open-pre- ijijij .

(ii) every issetopen--ij setopen-semi-,open-pre- ijij .

(iii) the concept of setsopen-semi-andopen-pre- ijij are independent .

2010

Proposition 2.6 :

A subset A of a bitopological space ),,( 21 X is setopen--ij if and only if there exists

an setopeni U , such that ))(int( UcljiAU .

Proof :

This follows directly from the definition (2.1) (iii) .

Proposition 2.7 : [Maheshwari , S . N . and Prasad , R . ,1977]

A subset A of a bitopological space ),,( 21 X is setopen-semi-ij if and only if there

exists an setopeni U , such that )(UcljAU .

Proposition 2.8 : [Jelic , M . , 1990]

A subset A of a bitopological space ),,( 21 X is setopen-pre-ij if and only if there

exists an setopeni U , such that )(AcljUA .

Theorem 2.9 :

A subset A of a bitopological space ),,( 21 X is an setopen--ij if and only if A is

setopen-pre-andsetopen-semi- ijij .

Proof :

Follows from definition (2.1) and remark (2.5) .

Definition 2.10 : [H . I. Al-Rubaye , Qaye , 2012]

Let ),,( 21 X be a bitopological space , A X . Then A is a said to be

open--semi- ij set if there exists an setopen--ij U in X , such that )(UcljAU .

The family of all open--semi- ij sets of X is denoted by )(- XOSij , where

2,1,; jiji .

Example 2.11 :

Let },,{ cbaX , }},{},{,,{1 baaX , and }},{},{},{,,{2 cbcbX .

),(and),( 21 XX are two topological spaces , then ),,( 21 X is a bitopological space .

The family of all setsopen--semi-21 of X is : }},{},{,,{)(-12 baaXXOS .


2011

The family of all setsopen--semi-12 of X is : }},{},{},{,,{)(-12 cbcbXXOS .

The following proposition will give an equivalent definition of open--semi- ij sets .

Proposition 2.12 : [H . I. Al-Rubaye , Qaye , 2012]

Let ),,( 21 X be a bitopological space , A X . Then A is an open--semi- ij set if

and only if ))))int((int(( AicljicljA .

Remark 2.13 : [H . I. Al-Rubaye , Qaye , 2012]The intersection of any two open--semi- ij sets is not necessary open--semi- ij set

as in the following example .

Example 2.14 :

Let },,{ cbaX , }},{},{},{,,{1 babaX , and }}{,,{2 aX . The family of all

setsopen--semi-21 of X is : }},{},,{},,{},{},{,,{)(-12 cbcababaXXOS .

Hence },{ ca and },{ cb are two setsopen--semi-21 , but }{},{},{ ccbca is not

setopen--semi-21 .

Proposition 2.15 : [H . I. Al-Rubaye , Qaye , 2012]

The union of any family of open--semi- ij sets is open--semi- ij set .

Remark 2.16 : [H . I. Al-Rubaye , Qaye , 2012]

(i) Every setopeni is open--semi- ij set ,but the converse need not be true .

(ii) If every setopeni is closedi and every nowhere setdensei is closedi in any

bitopological space , then every open--semi- ij set is an setopeni .

2012


(i) Every setopen--ij is open--semi- ij set , but the converse is not true in general .

(ii) If every setopeni is closedi set in any bitopological space , then every

open--semi- ij set is an setopen--ij .


The concepts of open--semi- ij and setsopen-pre-ij are independent , as the

following example .

Example 2.19 :

In example (2.3) , },{ cb is a setopen-pre-21 but not setopen--semi-21 .


(i) It is clear that every subsetsopen-pre-andopen-semi- ijij of any bitopological space is

open--semi- ij set ( by theorem (2.9) and remark (2.17) (i) ) .

(ii) An open--semi- ij set in any bitopological space ),,( 21 X is setopen-pre-ij if every

subsetopeni of X is closedi set ( from remark (2.17)(ii) and remark (2.5)(iii) ) .

Definition 2.21 : [H . I. Al-Rubaye , Qaye , 2012]

The complement of open--semi- ij set is called closed--semi- ij set. Then family of

all closed--semi- ij sets of X is denoted by )(- XCSij , where 2,1,; jiji .


The intersection of any family of closed--semi- ij sets is closed--semi- ij set .


2013


The following diagram shows the relations among the different types of weakly open

sets that were studied in this section :

3 . ij - Semi - - Compactness in Bitopological Spaces :

In this section the notion of compact--semi- ij space is introduced in bitopological

spaces and their relationships with compact--ij space and compact-pre-ij space are

studied .

+

+

+

open-pre-ij

open--ijopeni

open--semi- ij

open-semi-ij

closed-issetopen-every ii

closed-issetdense-nowhereevery

i

i

2014

Definition 3.1 :

Let ),,( 21 X be a bitopological space , A X , a family W of subsets of X is said to be

open--semi- ij cover of A if and only if W covers A and W is a subfamily of )(- XOSij .

Definition 3.2 :

A bitopological space ),,( 21 X is said to be compact--semi- ij if and only if every

open--semi- ij cover of X has a finite subcover .

Remark 3.3 :

Every compact--semi- ij space is compact-i .

Proof :

Follows from remark ( 2.16 ) (i) .

Remark 3.4 :

If every setopeni is closedi and every nowhere setdensei is closedi in any

bitopological space , then every compact-i space is compact--semi- ij .

Proof :

Follows from remark ( 2.16 ) (ii) .

Proposition 3.5 :

Every closed--semi- ij subset of compact--semi- ij space is compact--semi- ij .

Proof :

Let ),,( 21 X be compact--semi- ij space , and let A be closed--semi- ij subset of

X . Let }:{ UW be open--semi- ij cover of A .Since A is closed--semi- ij

subset of X, then AX is open--semi- ij subset of X . Therefore , the family

}{}:{ AXU is open--semi- ij cover of X , and since ),,( 21 X is

compact--semi- ij , it has a finite subcover . As A X and AX covers no part of A , a


2015

finite number of members of W , say nUUU ,........,, 21 , have the property that n

iiUA

1

.

Hence A is compact--semi- ij space .

Proposition 3.6 :

If A and B are two compact--semi- ij subsets of a bitopological space ),,( 21 X ,

then BA is compact--semi- ij .

Proof :

Follows from proposition ( 2.15 ) .

Definition 3.7 :

Let ),,(),,(: 2121 YXf be a function , then f is called :

(i) continuous--semi- ij if and only if the inverse image of each openi subset of Y is

open--semi- ij subset of X .

(ii) continuous--semi- *ij if and only if the inverse image of each open--semi- ij subset

of Y is open--semi- ij subset of X .

(iii) continuous--semi- **ij if and only if the inverse image of each open--semi- ij

subset of Y is an openi subset of X .

Remark 3.8 :

The following diagram explain the relationships among the different types of weakly

continuous function :

2016

continuous-i

continuous--semi- *ij

continuous--semi- **ij continuous--semi- ij

+onto,openi

Theorem 3.9 :

The continuous--semi- ij image of compact--semi- ij space is compact-i .

Proof :

Let ),,(),,(: 2121 YXf be an continuous--semi- ij , onto function and

let ),,( 21 X be compact--semi- ij space . To prove that Y is compact-i .

Let }:{ U be an openi cover of Y , implies }:)({ 1 Uf is open--semi- ij

cover of X , which is compact--semi- ij space . So , there exist n ,........,, 21 , such

that },.....,2,1:)({ 1 niUf i is a finite subcover of X . Implies },.....,2,1:{ niU i is a finite

subcover of Y . So Y is compact-i .

Corollary 3.10 :

The continuous-i image of compact--semi- ij space is compact-i .

Proof :Since every continuous-i function is continuous--semi- ij , theorem ( 3.9 ) is

applicable .


2017

Theorem 3.11 :

The continuous--semi- *ij image of compact--semi- ij space is

compact--semi- ij .

Proof :

Let ),,(),,(: 2121 YXf be an continuous--semi- *ij , onto function and let

),,( 21 X be compact--semi- ij space . To prove that ),,( 21 Y is compact--semi- ij .

Let }:{ U be open--semi- ij cover of Y , then }:)({ 1 Uf is open--semi- ij

cover of X , and since ),,( 21 X is compact--semi- ij space , then there exist an indices

n ,........,, 21 , such that },.....,2,1:)({ 1 niUf i is a finite subcover of X . Since f is

onto , },.....,2,1:{ niU i is a finite subcover of Y . So Y is compact--semi- ij .

Corollary 3.12 :

Let ),,(),,(: 2121 YXf be an openi , continuous-i function of X onto Y .

If ),,( 21 X is compact--semi- ij , then ),,( 21 Y is compact--semi- ij .

Proof :

Since every openi , onto , continuous-i function is continuous--semi- *ij . Theorem

(3.11) is applicable .

Corollary 3.13 :

Let }:{ X be any family of bitopological spaces . If the product space

X is

compact--semi- ij , then X is compact--semi- ij , for each .

Proof :

Since each projection

XX

: is continuous-i , openi function of

X

onto X . It follows , by corollary (3.12) , that each X is compact--semi- ij .

2018

Corollary 3.14 :

The continuous--semi- *ij image of compact--semi- ij space is compact-i .

Proof :

Follows from theorem (3.11) and remark (3.3) .

Theorem 3.15 :

The continuous--semi- **ij image of compact-i space is compact--semi- ij .

Proof :

Let ),,( 21 X be an compact-i space , and ),,(),,(: 2121 YXf be an

continuous--semi- **ij , onto function . To prove Y is compact--semi- ij space .

Let }:{ U be open--semi- ij cover of Y , then }:)({ 1 Uf is an openi cover

of X , and so can be reduced to a finite subcover )(..,),........(),( 12

11

1nUfUfUf

; it is

evident that nUUU ,........,, 21 is a finite subcover of Y . So Y is compact--semi- ij .

Corollary 3.16 :

The continuous--semi- **ij image of compact-i space is compact-i .

Proof :


Definition 3.17 :

Let ),,( 21 X be a bitopological space , A X , a family W of subsets of X is said to be

)open-pre-resp.(open-- ijij cover of A if and only if W covers A and W is a subfamily of

)(- XOij ( resp. )(- XPOij ) .

Definition 3.18 :

A bitopological space ),,( 21 X is said to be compact--ij )compact-pre-resp.( ij

space if and only if every open--ij )open-pre-resp.( ij cover of X has a finite subcover .


2019

Proposition 3.19 :

Every compact-pre-ij space is compact--ij .

Proof :

Follows from remark (2.5) (ii) .

Proposition 3.20 :

If every open-pre-ij set in a bitopological space ),,( 21 X is setopen-semi-ij , then X

is compact-pre-ij space , whenever it is an compact--ij space .

Proof :

Follows from theorem (2.9) .

Proposition 3.21 :

Every compact-pre-ij space is compact-i .

Proof :

Follows from remark (2.5) (i) .

Remark 3.22 :

Every compact--semi- ij space is compact--ij .

Proof :

Follows from remark (2.17) (i) .

Proposition 3.23 :

In a bitopological space ),,( 21 X , if every subsetopeni of X is closedi set , then

X is compact--semi- ij space , whenever it is an compact--ij space .

Proof :

Follows from remark (2.17) (ii) .

2020

Remark 3.24 :

The concepts of compact-pre-ij space and compact--semi- ij space are independent .

Proposition 3.25 :

If every setopeni in a bitopological space ),,( 21 X is closedi , then X is

compact--semi- ij , whenever it is compact-pre-ij .

Proof :

Follows from propositions (3.19) and (3.23) .

Proposition 3.26 :

If every open-pre-ij set in a bitopological space ),,( 21 X is setopen-semi-ij , then X

is compact-pre-ij space , whenever it is an compact--semi- ij space .

Proof :

Follows from remark (3.22) and proposition (3.23) .

Corollary 3.27 :

The continuous--semi- **ij image of compact-i space is compact--ij .

Proof :


Corollary 3.28 :

The continuous--semi- **ij image of compact--ij space is compact--semi- ij

( compact--ij , compact-i , respectively ) .

Proof :Clear .

Corollary 3.29 :

The continuous--semi- **ij image of compact--semi- ij space is compact--semi- ij

( compact--ij , compact-i , respectively ) .


2021

compact-pre-ij

compact--semi- ij

compact--ij compact-i

+

+

+

closed-issetopen-every ii

open-semi-issetopen-pre-every ijij

closed-issetdense-nowhereevery

i

i

Proof :

Clear .

Remark 3.30 :

The following diagram shows the relations among the different types of compactness :

2022

References :Bose , S. , 73 (1981) , “ Semi - open sets , semi - continuity and semi - open mappings in

bitopological spaces “ , Bull . Cal . Math . Soc ., 237 – 246 .

H . I. Al-Rubaye , Qaye , (2012) ,“ Semi - - Separation Axioms in Bitopological Spaces “ ,

AlMuthanna Journal of Pure Sciences , No.1, Vol.1 .

Jelic , M . , 3 (1990),“ A decomposition of pair wise continuity “, J . Inst . Math . Comput . Sci.

Math . Ser . , 25 – 29 .

Jelic , M . , 24 (1990) , “ Feebly p – continuous mappings “ , Suppl . Rend Circ . Mat . Palermo

(2), 387 – 395 .

Kar A . , BHATTACHARYYAP. , 34 (1992) , “ Bitopological pre open sets , pre continuity

and pre open mappings “ , Indian J . Math . , 295 – 309 .

Kelly, J .C ., 13 (1963) , “ Bitopological Spaces “, Proc . London Math . Soc . , 17 – 89 .

Kumar Sampath , S ., 89 (1997) , “ On a Decomposition of Pairwise Continuity “, Bull. Cal .

Math . Soc . , 441 – 446 .

Maheshwari , S . N . and Prasad , R . , 26 (1977/ 78 ) , “ Semi - open sets and semi -

continuous functions in bitopological spaces “ , Math . Notae , 29 – 37 .

Semi - α - Compactness in Bitopological Spaces

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