Semi - - Connectedness in Bitopological Spaces

Transcription

1 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second صفحة Sem - - Connectedness n Btopologcal Spaces By Qays Hatem Imran Al-Muthanna Unversty, College of Educaton, Department of Mathematcs Al-Muthann Iraq E-mal : alrubaye84@yahoocom Abstract : The objectve of ths paper s to study a specal case of connectedness n btopologcal spaces by consderng j -sem sets and ther relatonshps wth j - connected space and j - pre- connected space Key words : Btopologcal space, j - sem - - open set, j - sem- - connected space Introducton : The study of btopologcal spaces was ntated by Kelly, J C, [5] A trple X,, ) s called btopologcal space f X, ) and X, ) are two topologcal spaces In 997, Kumar Sampath, S, [6] ntroduced the concept of Bose, S, [] ntroduced the noton of A, [4] have ntroduced the noton of j setsn btopologcal spaces In 98, j - sem - open sets n btopologcal spaces In 99, Kar j - pre- open sets n btopologcal spaces In 0, H I Al-Rubaye, Qaye, [] ntroduced the noton of j -sem sets n btopologcal spaces In ths paper, we study especal case of connectedness n btopologcal spaces by consderng j -sem sets, we prove some results about them comparng wth smlar cases n topologcal spaces Prelmnares : Throughout the paper, spaces always mean a btopologcal spaces, the closure and the nteror of any subset A of X wth respect to, wll be denoted by respectvely, for, A, and nt A cl 35

2 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second Defnton : ) ) Let X,, ) be a btopologcal space, A X, A s sad to be : j - pre set [4] f A nt j cl ), where j;, j,, j -sem set [] f A j cl nt ), where j;, j,, ) j set [6] f A nt j cl nt )), where j;, j, Remark : The famly of j - pre- open resp j -sem and j ) sets of X s denoted by j - PO X ) resp j - SO X ) and j O X ) ), where j;, j, Example 3 : Let X {, { X, }}, and { X,,{ }} a c X, ) and X, ) are two topologcal spaces, then X,, ) s a btopologcal space The famly of all - pre sets of X s : - PO X ) { X, } The famly of all The famly of all -sem sets of X s : - SO X ) { X, a}} sets of X s : O X ) { X, a}} Defnton 4 : The complement of an j - pre- open resp j -sem and j ) set s sad to be j - pre- closed resp j -sem and j ) set The famly of j - pre- closed resp j -sem and j ) sets of X s denoted by j - PC X ) resp j - SC X ) and j C X ) ), where j;, j, Remark 5 : It s clear by defnton that n any btopologcal space the followng hold : ) every open set s j - pre, j -sem, j set ) every j set s j - pre, j -sem set ) the concept of j - pre- open and j -sem sets are ndependent Proposton 6 : A subset A of a btopologcal space X,, ) s j set f and only f there exsts an open set U, such that U A nt j cl U)) Ths follows drectly from the defnton ) ) 36

3 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second Proposton 7 : [7] A subset A of a btopologcal space X,, ) s j -sem set f and only f there exsts an open set U, such that U A j clu ) Proposton 8 : [3] A subset A of a btopologcal space X,, ) s j - pre set f and only f there exsts an open set U, such that A U j cl Theorem 9 : A subset A of a btopologcal space X,, ) s an j set f and only f A s j -sem set and j - pre set Follows from defnton ) and remark 5) Defnton 0 : [4,6] Let X,, ) be a btopologcal space and A X,the ntersecton of all j - closed resp j - pre- closed ) sets contanng A s called j - closure resp j - pre- closure ) of A, and s denoted by j - cl resp j - pre- cl ) ; e j - cl { B X : B s j - closed set, A X} and j - pre- cl { B X : B s j - pre- closed set, A X} Defnton : [] Let X,, ) be a btopologcal space, A X Then A s a sad to be j -sem set f there exsts an j set U n X, such that U A j clu ) The famly of all j -sem sets of X s denoted by j - S O X ), where j;, j, The followng proposton wll gve an equvalent defnton of Proposton : [] j -sem sets Let X,, ) be a btopologcal space, A X Then A s an j -sem set f and only f A j cl nt j cl nt ))) Remark 3 : [] The ntersecton of any two n the followng example j -sem sets s not necessary j -sem set as 37

4 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second Example 4 : Let X {, { X, }}, and { X, }} The famly of all b a -sem sets of X s : - SO X ) { X,,{ } Hence { and { are two -sem sets, but { { { s not -sem set Proposton 5 : [] The unon of any famly of j -sem sets s j -sem set Remark 6 : [] ) open set s j -sem set,but the converse need not be true ) If every open set s closed and every nowhere dense set s closed n any btopologcal space, then every j -sem set s an open set Remark 7 : [] ) j set s j - sem - open set, but the converse s not true n general ) If every open set s closed set n any btopologcal space, then every j -sem set s an j set Remark 8 : [] The concepts of j -sem and j - pre sets are ndependent, as the followng example Example 9 : In example 3), { s a- pre set but not -sem set Remark 0 : [] ) It s clear that every j -sem and j - pre subsets of any btopologcal space s j -sem set by theorem 9) and remark 7) ) ) 38

5 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second ) An j -sem set n any btopologcal space X,, ) s j - pre set f every open subset of X s closed set from remark 7) ) and remark 5) ) ) Defnton : [] The complement of j -sem set s called j -sem set Then famly of all j -sem sets of X s denoted by j - S C X ), where j;, j, Remark : [] The ntersecton of any famly of j -sem sets s j -sem set Defnton 37 : [] Let X,, ) be a btopologcal space and A X,the ntersecton of all j -sem sets contanng A s called j -sem -closure of A, and s denoted by j - S - cl ; e j - S - cl { B X : B s j - sem - closed set, A X} Remark 3 : [] The followng dagram shows the relatons among the dfferent types of weakly open sets that were studed n ths secton : j - pre j -sem open j every nowhere - dense set s - closed every - open set s j -sem 39

6 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second 3 j - Sem - - Connectedness n Btopologcal Spaces : In ths secton the noton of spaces and ther relatonshps wth j - sem - connected space s ntroduced n btopologcal j - connected space and j - pre- connected space are studed Defnton 3 : Let X,, ) be a btopologcal space, two non-empty subsets A and B of X are sad to be j - sem - separated f A j - S - cl B) and j - S - cl B Defnton 3 : A btopologcal space X,, ) s called j - sem - connected f t s not the unon of two non-empty j - sem - separated j -sem sets A subset B X s j - sem - connected f t s j - sem - connected as a subspace of X An j - sem - dsconnect on of X s a par of complement, non-empty, j -sem j - sem - closed subsets Remark 33 : The only j -sem j - sem - closed subsets n j - sem - connected space X are X and Remark 34 : j - sem - connected space s - connected, but the converse s not true and Suppose that X s not - connected, then two non-empty A, B are open A B A B X By remark 6 ) ), we have A, B are j -sem sets, A B X and A B, hence X s not j - sem - connected whch s a contradcton Thus, X s - connected But the converse s not true as n the followng example 40

7 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second Example 35 : Let X {, { X, }}, { X, }} b c The famly of all -sem sets of X s : - SO X ) { X, } Then X s - connected space, but X s not -sem - connected Remark 36 : If every open set s closed and every nowhere dense set s closed n any btopologcal space, then every - connected space s j - sem - connected Follows from remark 6 ) ) Defnton 37 : A functon f X,, ) Y,, ) s called j - sem - open f for each open set : U of X, f U ) s j - sem - open n Y Defnton 38 : A functon f X,, ) Y,, ) s called j - sem - contnuous f and only f the nverse mage of each Proposton 39 : : open subset of Y s j -sem subset of X contnuous functon s j - sem - contnuous Follows from remark 6 ) ) Defnton 30 : A functon f X,, ) Y,, ) s called j - sem - rresolute f and only f the nverse mage of each Proposton 3 : : j -sem subset of Y s j -sem subset of X j - sem - rresolute functon s j - sem - contnuous 4

8 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second Let A be any open set n Y Then we have A s an j -sem set n Y [ from remark 6) ) ] Snce f s j - sem - rresolute functon,then f A ) s j -sem set n X Therefore f s j - sem - contnuous Proposton 3 : and f s Let X,, ) and Y,, ) be two btopologcal spaces If X s j - sem - connected - connected j - sem - contnuous functon from X,, ) onto Y,, ), then Y s Suppose that A s an open closed subset of Y, then f A ) s j -sem j - sem - closed n X Hence f A ) s or X, but X s j - sem - connected So A s or Y Hence Y s Proposton 33 : - connected An j - sem - rresolute mage of any j - sem - connected btopologcal space s j - sem - connected Follows drectly from proposton 3 ) Defnton 34 : Let X,, ) be a btopologcal space, two non-empty subsets A and B of X are sad to be j - separated resp j - pre- separated ) f A j - cl B) resp A j - pre- cl B) ) and j - - cl B resp j - pre- cl B ) Defnton 35 : A btopologcal space X,, ) s called j - connected resp j - pre- connected ) space f t s not the unon of two non-empty j - separated resp j - pre- separated ) j - open resp j - pre- open ) sets 4

9 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second Proposton 36 : j - pre- connected space s j - connected Follows from remark 5) ) Proposton 37 : If every j - pre- open set n a btopologcal space X,, ) s j -sem set j - pre- connected space, whenever t s an j - connected space, then X s Follows from theorem 9) Proposton 38 : j - pre- connected space s - connected Follows from remark 5) ) Remark 39 : j - sem - connected space s j - connected Follows from remark 7) ) Proposton 30 : s In a btopologcal space X,, ), f every open subset of X s closed set, then X j - sem - connected space, whenever t s an j - connected space Follows from remark 7) ) Remark 3 : The concepts of j - pre- connected space and j - sem - connected space are ndependent 43

10 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second Proposton 3 : If every open set n a btopologcal space X,, ) s closed, then X s j - sem - connected, whenever t s j - pre- connected Follows from propostons 36) and 30) Proposton 33 : If every j - pre- open set n a btopologcal space X,, ) s j -sem set, then X s j - pre- connected space, whenever t s an j - sem - connected space Follows from remark 39) and proposton 30) Remark 34 : The followng dagram shows the relatons among the dfferent types of connectedness : j - pre- connected every j - pre set s j -sem j - connected - connected every nowhere - dense set s - closed every - open set s j - sem - connected 44

11 Journal of AL-Qadsyah for computer scence an mathematcs A specal Issue Researches of the fourth Internatonal scentfc Conference/Second References : [] Bose, S, Sem - open sets, sem - contnuty and sem - open mappngs n btopologcal spaces, Bull Cal Math Soc, 73 98), [] H I Al-Rubaye, Qaye, Sem - - Separaton Axoms n Btopologcal Spaces,AL- Muthanna Journal of Pure Scences, No, Vol, September, 0), [3] Jelc, M, A decomposton of par wse contnuty, J Inst Math Comput Sc Math Ser, 3 990), 5 9 [4] Kar A, Bhattacharyya P, Btopologcal pre open sets, pre contnuty and pre open mappngs, Indan J Math, 34 99), [5] Kelly, J C, Btopologcal Spaces, Proc London Math Soc, 3 963), 7 89 [6] Kumar Sampath, S, On a Decomposton of Parwse Contnuty, Bull Cal Math Soc, ), [7] Maheshwar, S N and Prasad, R, Sem - open sets and sem - contnuous functons n btopologcal spaces, Math Notae 6 977/ 78 ),