## Bifuzzy topological spaces

Abu Safiya, A. S. M. (1993).
*Bifuzzy topological spaces*.
(Unpublished Doctoral thesis, City, University of London)

## Abstract

Separation axioms for bifuzzy topological spaces namely : P-Ri, P-Tj,P-Tjw (i=l,2,j=0,l,2,2 l/2),P-regular and P-normal spaces are defined and many related results are proved such as a bfts (X,τ1, τ2) Is P-normal iff for every τi-closed fuzzy set λ and τj-open 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 S-connected,Sw-connected ,P-connected and Pw-connected .We have shown that connectedness is preserved under P-continuity 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 S-compact, P-compact, S-α-compact, S-weakly compact, S-α-weakly compact, P-weakly compact, P-α-weakly compact, S-C- compact, P-C-compact, S-C weakly compact, P-C-weakly compact, P-U- compact and P-S-compact .We have proved that P-S-compactness => P-C-compactness => P-U-compactness but P-U-compactness does not imply neither P-C-compactness nor P-S-compactness. 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 S-weakly compact and S-C- compact in Lindelof spaces. Finally we introduce induced and weakly induced bifuzzy topological spaces and prove that a P-Hausdorff compact bfts is P-weakly induced and a P-topological P-weakly induced bfts is P-induced. Lowen's goodness criterion is extended and then used to test the goodness of these definitions. We have proved that (X,T1,T2) is P-Ti, P-Tiw,P-regular and P-normal iff the bifuzzy topological space (X,ω(T1), ω(T2)) is P-Ti,P-Tiw (i=0,l,2,2 1/2), P- regular and P-normal respectively. We have shown that S- connectedness, P-connectedness are good extensions while Sw- connectedness and Pw-connectedness are not. Moreover we have also shown that S-α-compactness is a good extension of S-compactness if it is good for some α∈ [0,l); while P-α-compactness is a good extension of P-compactness only for α=0. Finally we prove a bitopological space (X,T1,T2) has P-f.p.p iff (X,ω(T1),ω(T2)) has P-f.p.p .

[Please see inside the thesis for a better view of equations]

Publication Type: | Thesis (Doctoral) |
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Subjects: | Q Science > QA Mathematics |

Departments: | Doctoral Theses School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses |

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