Title:

Bifuzzy topological spaces

Separation axioms for bifuzzy topological spaces namely : PRi, PTj,PTjw (i=l,2,j=0,l,2,2 l/2),Pregular and Pnormal spaces are defined and many related results are proved such as a bfts (X,τ1, τ2) Is Pnormal iff for every τiclosed fuzzy set λ and τjopen fuzzy set μ such that λ C μ there exists a continuous function f : (X,τ1,τ2)>([0,1]f, L,R) such that λ(x)≤f(x)(l )≤ f(x)(0+)≤μ(x), for all x ∈ X.Bifuzzy connected topological spaces are defined such as Sconnected,Swconnected ,Pconnected and Pwconnected .We have shown that connectedness is preserved under Pcontinuity and we have shown that the connectedness of (X,τ1,τ2) is not governed by the connectedness of (X, τ1) and (X, τ2). Many types of compactness were defined such as Scompact, Pcompact, Sαcompact, Sweakly compact, Sαweakly compact, Pweakly compact, Pαweakly compact, SC compact, PCcompact, SC weakly compact, PCweakly compact, PU compact and PScompact .We have proved that PScompactness => PCcompactness => PUcompactness but PUcompactness does not imply neither PCcompactness nor PScompactness. Also we have shown that bifuzzy compactness is preserved under continuous surjection. Bifuzzy Lindelof spaces are also defined. We have shown that there are no analogous definitions of Sweakly compact and SC compact in Lindelof spaces. Finally we introduce induced and weakly induced bifuzzy topological spaces and prove that a PHausdorff compact bfts is Pweakly induced and a Ptopological Pweakly induced bfts is Pinduced. Lowen's goodness criterion is extended and then used to test the goodness of these definitions. We have proved that (X,T1,T2) is PTi, PTiw,Pregular and Pnormal iff the bifuzzy topological space (X,ω(T1), ω(T2)) is PTi,PTiw (i=0,l,2,2 1/2), P regular and Pnormal respectively. We have shown that S connectedness, Pconnectedness are good extensions while Sw connectedness and Pwconnectedness are not. Moreover we have also shown that Sαcompactness is a good extension of Scompactness if it is good for some α∈ [0,l); while Pαcompactness is a good extension of Pcompactness only for α=0. Finally we prove a bitopological space (X,T1,T2) has Pf.p.p iff (X,ω(T1),ω(T2)) has Pf.p.p.
