On Some Covering Properties of B-open sets

Transcription

1 O Some Coverg Propertes of B-ope sets Belal k Narat Appled Sceces Prvate verst Amma-Jorda Astract I ths paper we troduce ad stud the cocepts of -ope set, -cotuous fuctos, the we also stud the cocepts of - compact susets ad stud some ew characterzatos of - separato aoms such as - T 2 The we dscuss the relatos etwee the -cotuous fuctos ad these cocepts Kewords -ope set, -compact, -ope cover, -closed sets, -cotuous 2 Itroducto Geeralzed ope sets pla a ver mportat role Geeral Topolog ad the are ow the research topcs of ma topologsts worldwde Leve [7] troduced the oto of sem-ope sets ad sem-cotut topologcal spaces Adrjevc [ 2] troduced a class of geeralzed ope sets topologcal spaces Mashhour [9] troduced pre ope sets topologcal spacesthe class of -ope sets s cotaed the class of sem-ope ad pre-ope sets ths paper we dscuss the coverg propertes of - sets ad - cotuous fuctos All through ths paper(,τ ) ad (Y,σ )stad for topologcal spaces wth o separato assumed, uless otherwse stated the closure of A ad the teror of A wll e deoted Cl(A) ad It(A), respectvel 3 Prelmares Defto 3 A suset A of a space s sad to e [2],[0]: Sem-ope f A Cl(It(A)) 2 Pre ope f A It(Cl(A)) 3 -ope f A It(Cl(It(A))) 4 β -ope f A Cl(It(Cl(A))) 5 -ope f A Cl(It(A)) It(Cl(A)) 48

2 Defto 32 A fucto f : Y s called [], [9]: sem cotuous f V s sem ope for each ope set V of Y 2 pre cotuous f V s pre ope for each ope set V of Y 3 -cotuous f V s - ope for each ope set V of Y 4 β -cotuous f V s β -ope for each ope set V of Y 5 -cotuous f V s -ope for each ope set V of Y Defto 33 [0] A space s a -T 2 space ff for each, such that there are -ope sets, V so that, V ad V 4 Coverg Propertes Defto 4 Let { G } e a faml of -ope sets of the space the faml { } : G : covers f G Defto 42 A space s called a -compact space f each -ope cover of has a fte sucover for Theorem 43 Let e a -compact suset of the -T 2 space ad the there est two dsjot -ope sets ad cotag ad, respectvel Proof : Let, sce s -T 2 space there est two -ope sets, V such that, V, IV φ, the faml { AIV : A} s ope cover of has a fte sucover { AI V AIV, K, AIV }, 2, thus K 2 49

3 Theorem 44 If s -T 2 space ad A s a -ope suset, f A s -compact the A s a -closed Let, the theorem 43 there est two -ope sets ad such that V A,whch mples s -ope so that A s -closed, A V, I V φ, thus Theorem 45 Let ad e a two -compact susets of the -T 2 space, the there est dsjot -ope sets ad cotag ad, receptvel Let, sce A s a -compact suset ad -ope, there est two -ope sets, V such that I V φ ; V, A, so β { BIV ; B} s a -ope cover of B, sce B s -compact suset there est fte sucover { B V ; } I from β I Let, V V, thus A, B V, I V φ Theorem 45 Let : (,τ ) ( Y,ρ ) compact f e a cotuous surjecto ope fucto, f s a -compact the Y s a - { V : } Let β { V : } e a -ope cover of Y, the f ( ) -compact space, there est a fte sucover from L to the space such that L s a -ope cover of sce s a f ( V ), thus Y f ( ) f f ( V ) f f ( V ) ( V ) Hece Y ( V ), ths shows Y s a -compact 50

4 Corollar 46 B-compactess s a topologcal propert The proof from theorem Theorem 45 Defto 47: A faml of sets β has fte tersecto propert f ever fte sufaml of β has a oempt tersecto Theorem 45 A topologcal space s compact f ad ol f a collecto of ts closed sets havg the fte tersecto propert has o-empt tersecto Suppose s -compact, e, a collecto of -ope susets that cover has a fte collecto that also cover G s a artrar collecto of -closed susets wth the fte tersecto propert We Further, suppose { } : clam that IG φ s o-empt Suppose otherwse, e, supposeig φ The ( G ) IG φ Sce each G s -closed, the collecto { G : } cover for B compactess, there s a fte sucover L such that ( G ) But the IG ( ( G ) ( G ) φ I tersecto propert of { G : } s a -ope, whch cotradcts the fte Coversel, take the hpothess that ever faml of a -closed sets havg the fte tersecto propert has a oempt tersecto we are to show s compact let{ G : } e a -ope cover of the { G : } s a faml of -closed sets such that I G G φ Cosequetl, our hpothess mples the faml { G : } does ot have the fte tersecto propert Therefore, there s some fte su collecto { G :,2,3, K, } I such that G φ ad hece G ( ( G ) ( G ) φ I Thus G, mplg s -compact 5

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