Separation Axioms of Fuzzy Bitopological Spaces

Transcription

1 IJCSNS Internatonal Journal of Computer Scence and Network Securty VOL3 No October 3 Separaton Axom of Fuzzy Btopologcal Space Hong Wang College of Scence Southwet Unverty of Scence and Technology Manyang 6 Schuan PRChna Abtract In th paper we gve and tudy dfferent type of eparaton axom ung the remoted neghbourhood of a fuzzy pont and a fuzzy et n the fuzzy upratopologcal Space ( τ whch generated by the fuzzy btopologcal pace ( properte on thee eparaton axom are reearched Keyword: τ τ Several Fuzzy upratopologcal fuzzy btopologcal remoted neghbourhood eparaton axom Introducton ASMahhoue FHKehdr and MHChenm [3] defned the fuzzy btopologcal pace ( τ τ AKandl ADNouh and SA Shekh [4] defned the fuzzy upratopologcal Space ( τ generated by the fuzzy btopologcal pace ( τ τ CKwong [7] ntroduced the concept of fuzzy pont and ther neghourhood But there are ome drawback n th tudy For overcomng the problem that tradtonal neghbourhood method wa no longer effectve n fuzzy topology Lu and Pu [6] ntroduced the concept of the o-called - neghbourhood Nearly at the ame tme Wang [5] ntroduced the concept of the o-called remoted neghbourhood to tudy fuzzy topology The latter concept ha more extenve applcaton than the former one Nhmura [8] defned the trong neghbourhood of the fuzzy pont and he defned the fuzzy flter generated by all the open neghbourhood and opened trong neghbourhood of the fuzzy pont In ecton 3-6 we defne two type of -pace and FPR-pace and tudy ome properte on them Notaton and Prelmnare All fuzzy et on unvere wll be denoted by I The cla of all fuzzy pont n unvere wll be denoted by FP( Ue Greek letter a µηδ etc to denote fuzzy et on Alo P tand for parwe A fuzzy pont [5] p σ be defned a the ordered par ( p σ ( I {} where I = [] If σ µ ( p then ( p σ µ and we call ( p σ belong to µ Alo f σ < µ ( p then ( p σ µ and we call ( p σ belong trongly to µ Defnton[3] Let be any et andτ τ be two fuzzy topologe on The trple ( τ τ ad to be a fuzzy btopologcal pace Defnton[8] Let ( τ be an fuzzy topologcal pace and ( p σ FP( Then ( An fuzzy et µ t ( p σ µ ad to be an open neghbourhood of ( p σ The fuzzy flter generated by all the open neghbourhood of ( p σ denoted and defned a : V( p σ = { µ I : ρ τ( p σ ρ µ } Each fuzzy et belongng to V ( p σ ad to be an neghbourhood of ( p σ ( An open fuzzy et µ t ( p σ µ ad to be an open -neghbourhood of ( p σ The fuzzy flter generated by all the open -neghbourhood of ( p σ denoted and defned a : V( p σ = { µ I : ρ τ( p σ ρ µ } Each fuzzy et belongng to V ( p σ ad to be an -neghbourhood of ( p σ Manucrpt receved October 5 3 Manucrpt reved October 3

2 IJCSNS Internatonal Journal of Computer Scence and Network Securty VOL3 No October 3 Defnton3[6] The fuzzy flter generated by all the open -neghbourhood of ( p σ denoted and defned a : V( p σ = { µ I : ρ τ( p σ qρ µ } Each fuzzy et belongng to V ( p σ ad to be an -neghbourhood of ( p σ Defnton4[6] Let be a crp ubet of an fuzzy topologcal pace ( τ Then ( τ ad to be a ubpace of (τ whereτ a fuzzy topology on gven by τ = { µµ τ} A ubpace ( τ open (cloed f the crp fuzzy et open (cloed nτ Defnton5[9] Let ( τ be an fuzzy topologcal pace and µρ I A fuzzy et ρ ad to be: ( R- neghbourhood of a fuzzy pont ( p σ f for ome cloed fuzzy et λ have ( p σ λ ρ ( R - neghbourhood of a fuzzy pont ( p σ f for ome cloed fuzzy et λ have ( p σ λ ρ The collecton of all the R- neghbourhood of ( p σ (rep R - neghbourhood of ( p σ denoted by R( p σ (rep R( p σ e R( p σ = { ρ I : λ τ( p σ λλ ρ} ( R( p σ = { ρ I : λ τ( p σ λλ ρ} Defnton6[4] Let ( τ τ be fuzzy btopologcal pace and µ I Then:Aocated wth the fuzzy cloure operatorτ cl and τ cl defne the mappng : C I I a: C( µ = τ cl ( µ τ cl ( µ Defnton7[4] Let ( τ τ be fuzzy btopologcal pace Then : the par ( τ ad to be the aocated fuzzy upratopologcal pace of ( τ τ where τ = { µ I : µ = µ µ µ τ µ τ} µ τ ad to be fuzzy τ -open or fuzzy upraopen n ( τ τ and t complement ad to be fuzzy uprecloed n ( τ τ Theorem[9] Let (τ be an fuzzy topologcal pace and λ I Then : ( V( p σ = V ( λ V ( p σ ( p σ ff λ R( p σ ( λ R( p σ ff λ V( p σ (v λ V( p σ ff R( p σ λ Theorem[9] If (τ be an fuzzy topologcal pace and ( τ a ubpace of (τ µ I We defne: ( R( p σ = { ρρ R( p σ } ( R( p σ = { ρρ R( p σ } Then R( p σ (rep R ( p σ the collecton of R- neghbourhood (rep R - neghbourhood of µ n the pace ( τ 3 Separaton axom FP T and Defnton3 An fuzzy btopologcal pace ( τ τ ad to be ( FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -cloed et µ (rep µ τ τ t ( µ R( p σ µ R or ( ( q δ µ R( q δ µ R ( p σ ( FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -cloed et µ (rep µ τ τ t ( µ R( p σ µ R or ( ( q δ µ R( q δ µ R ( p σ Theorem3 Let ( τ τ be fuzzy btopologcal pace Then : ( ( τ τ FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et λ (rep λ τ τ t ( λ V( p σ λ V or ( ( q δ λ V( q δ λ V ( p σ Iff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et λ (rep λ τ τ t ( λ V( p σ λ V or ( ( q δ λ V( q δ λ V ( p σ (( τ τ FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et λ (rep λ τ τ t ( λ V( p σ λ V or ( ( q δ λ V( q δ λ V ( p σ Proof ( Follow from Theorem ((v and by puttng µ = λ n Defnton 3( ( Follow from Theorem ( and by puttng µ = λ n Defnton 3( Theorem3 Let ( τ τ be fuzzy btopologcal pace Then : ( FP T ( FP T

3 IJCSNS Internatonal Journal of Computer Scence and Network Securty VOL3 No October 3 3 Proof We prove part (and proof of the other part mlar Suppoe ( τ τ FP T ( q δ FP( p q Snce ( τ τ µ τ t ( µ R( p σ µ R( q δ Butτ τ τ then µ τ τ or ( µ R( q µ R( p Let ( p σ FP T then or ( µ R( µ R ( q δ p σ t ( µ R( p σ µ R ( q δ Hence ( τ τ δ σ Theorem33 A ubpace of a (rep (rep Proof We prove the cae and the proof of the other cae mlar Suppoe ( τ τ a ubpace of a FP T Let ( p σ ( q δ FP( p q Then ( p σ ( q δ FP( Snce ( τ τ then µ τ τ t ( µ R( p σ µ R or ( q δ ( µ R( q δ µ R So ( p σ µ = µ τ τ t ( µ R( p σ µ R or ( ( q δ µ R( q δ µ R ( p σ where R( p σ = { ρ: ρ R( p σ } Hence ( τ τ Theorem34 A ubpace of a FP T (rep FP T FP T (rep FP T Proof It mlar to that of Theorem33 4 Separaton axom FP T and Defnton4 An fuzzy btopologcal pace ( τ τ ad to be ( FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -cloed et µλ (rep µλ τ τ t ( µ R( p σ µ R and ( ( q δ λ R( q δ λ R ( p σ ( FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -cloed et µλ(rep µλ τ τ t( µ R( p σ µ R and ( q δ λ R λ R ( ( q δ ( p σ Theorem4 Let ( τ τ be fuzzy btopologcal pace Then : ( ( τ τ FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et ηρ (rep ηρ τ τ t η V p σ η V and ( q δ ρ V( q δ ρ V ( p σ ( ( ( Iff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et ηρ (rep ηρ τ τ t ( η V( p σ η V and( ( q δ ρ V( q δ ρ V ( p σ (( τ τ FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et ηρ (rep ηρ τ τ t ( η V( p σ η V and ( ( q δ ρ V( q δ ρ V ( p σ Proof ( Follow from Theorem ((v and by puttng µ = η λ = ρ n Defnton4( ( Follow from Theorem ( and by puttng µ = η λ = ρ n Defnton4( Theorem4 Let ( τ τ be fuzzy btopologcal pace Then : ( FP T ( FP T Proof It follow from the face that τ τ τ Theorem43 Let ( τ τ be fuzzy btopologcal pace Then :( FP T (rep FP T (rep ( FP T (rep FP T (rep Proof Obvou Theorem44 A ubpace of a (rep (rep Proof We prove the cae and the proof of the other cae mlar Suppoe ( τ τ a ubpace of a Let ( p σ ( q δ FP( p q Then ( p σ ( q δ FP( Snce ( τ τ then: µλ τ τ t ( µ R( p σ µ R and ( q δ ( λ R( q δ λ R So ( p σ µ = µ τ τ and λ = λ τ τ t( µ R( p σ µ R and ( q δ ( λ R λ R Hence ( τ τ ( q δ ( p σ Theorem45 A ubpace of a FP T (rep FP T FP T (rep FP T Proof It mlar to that of Theorem44 5 Separaton axom FP T and Defnton5 An fuzzy btopologcal pace ( τ τ ad to be ( FP T (rep ff ( p σ

4 4 IJCSNS Internatonal Journal of Computer Scence and Network Securty VOL3 No October 3 ( q δ FP( p q mple that there ext a fuzzy τ -cloed et µλ (rep µλ τ τ where µ R( p σ λ R t ( q δ λ µ = ( FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -cloed et µλ(rep µλ τ τ where µ R( p σ λ R( q δ t λ µ = Theorem5 Let ( τ τ be fuzzy btopologcal pace Then : ( ( τ τ FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et ηρ (rep ηρ τ τ where η V( p σ ρ V and ( q δ η ρ = Iff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et ηρ (rep ηρ τ τ where η V( p σ ρ V and ( q δ δ ρ = ( ( τ τ FP T (rep ff ( p σ ( q δ FP( p q mple that there ext a fuzzy τ -open et ηρ (rep ηρ τ τ where η V( p σ ρ V and ( q δ η ρ = Proof It mlar to that of Theorem4 Theorem5 Let ( τ τ be fuzzy btopologcal pace Then : ( FP T ( FP T Proof It follow from the face that τ τ τ Theorem53 Let ( τ τ be fuzzy btopologcal pace Then :( FP T (rep FP T (rep ( FP T (rep FP T (rep Proof We prove the cae ( and the proof of the other cae mlar Suppoe ( τ τ Let ( p σ ( q δ FP( p q then: µλ τ τ where µ R( p σ λ R t ( q δ λ µ = Snce µ R( p σ then ( p σ µ and nce λ µ = then ( p σ λ So λ R( p σ Smlar we have µ R( q δ Hence ( τ τ Theorem54 A ubpace of a (rep (rep Proof We prove the cae and the proof of the other cae mlar Suppoe ( τ τ a ubpace of a Let ( p σ ( q δ FP( p q Then ( p σ ( q δ FP( Snce ( τ τ then: µλ τ τ where µ R( p σ λ R( q δ t λ µ = So µ = µ τ τ and λ = λ τ τ t λ µ = Hence ( τ τ Theorem55 A ubpace of a FP T (rep FP T FP T (rep FP T Proof It mlar to that of Theorem54 6 Separaton axom FP R and FPR Defnton6 An fuzzy btopologcal pace ( τ τ ad to be ( FP R (rep FPR ff τ -cl( ( p σ qµ = (rep C (( p σ qµ µ R ( p σ ( FP R (rep FPR ff τ -cl( ( p σ qµ = (rep C (( p σ qµ µ R( p Theorem6 Let ( τ τ be fuzzy btopologcal pace Then : ( ( τ τ FP R (rep FPR ff ( p σ λ = (rep C (( p σ λ λ V Iff ( p σ ( p σ λ = (rep C (( p σ λ λ V( p σ ( ( τ τ FP R (rep FPR ff ( p σ λ = (rep C (( p σ λ λ V( p σ Proof It mlar to that of Theorem3 Theorem6 Let ( τ τ be fuzzy btopologcal pace Then : ( FP R FPR ( FP R FPR Proof We prove part (and proof of the other part mlar Suppoe ( τ τ FP R Let µ R ( p σ thenτ -cl( ( p σ qµ = ThuC (( p σ qµ Hence ( τ τ FPR Theorem63 A ubpace of a FPR (rep FPR FPR (rep FPR Proof We prove the cae FPR and the proof of the other cae mlar Suppoe ( τ τ a ubpace of σ

5 IJCSNS Internatonal Journal of Computer Scence and Network Securty VOL3 No October 3 5 an FPR Let µ R then ( p σ µ R Snce ( p σ ( τ τ FPR then C (( p σ qµ So C (( p σ qµ where C (( p σ = C (( p σ and R = { ρ: ρ R } Hence ( τ τ FPR ( p σ ( p σ Theorem64 A ubpace of a FP R (rep FP R FP R (rep FP R Proof It mlar to that of Theorem63 Smlar wth 4 and 5 we can further tudyseparaton axom FP R and FPR (rep FP R and FPR Omtted here Reference [] BHutton Normalty n fuzzy topologcal pace JMath Anal Appl 5 ( [] MEAbd El-Monef and AFRamadanOn fuzzy upratopologcal pace Indan J Pure and Appl Math8 (4 ( [3] ASMahhour and MHGhanm Fuzzy cloure pace JMath Anal Appl 6 ( [4] AKandl AANouh and SAShekh on fuzzy btopologcal pace fuzzy et and ytem 6(5 ( [5] GJWang A new fuzzy compactne defned by fuzzy net JMath Anal Appl 94 (983-4 [6] PPMng and LMng Fuzzy topology JMath Anal Appl 76 ( [7] CKWong Fuzzy pont and local porperte of fuzzy topology JMath Anal Appl 56 ( [8] TNhmura Fuzzy Monad Theory and Fuzzy Topology Math Japonce 3 ( [9] H Wang On Weakly Compact Set n L-Fuzzy Topologcal Space Internatonal Journal of Computer Scence and Network Securty (6 P [] H Wang The Refnement of Fuzzy Topogenou tructure and It Properte Fuzzy Sytem and Math 5(4 ( 83-86