الفهرس 
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Abstract
Zadeh, in 1965, introduced the concept of the fuzzy subset. Subsequently Chang defined the fuzzy topology in 1968. Moreover, many mathematicians have created interesting mathematical structures and predominant results in fuzzy topology. In 1999, Dib had studied the relation between the fuzzy topology on the family of P^* (I)-fuzzy subsets P^* (I)^X induced by a given topology on I^X , where I=[0,1] is the closed unit interval of real numbers and P^* (I) is the subset of the power set P(I), containing the zero element. Additionally, he found interesting relations and important properties between them. This stimulates special interest to conduct further elucidating research studies.
In this thesis, the definition of the fuzzy functions is reformulated to be applied to the (L,M)-fuzzy topological spaces. The 〖(P〗^* (L),M)-fuzzy topological spaces are studied to find out the common and the disparate properties with the (L,M)-fuzzy topological space. The extension of the proximity on the fuzzy space L^Y to a proximity on the fuzzy space L^X; Y⊂X and the restriction of the fuzzy proximity on L^X to a fuzzy proximity on L^Y are defined and studied, together with the relations between their closure operators. The induced basic proximity on P(Λ)^X for each given basic proximity on L^X is also defined where L∈L(Λ) and L(Λ) is the family of all complete lattices defined on the nonempty set Λ and fundamental relations between their closure operators are obtained. Furthermore, the (L,M)-fuzzy proximity is approached. Also, the restriction and the extension of (L,M)-fuzzy proximities and the induced (L,M)-fuzzy proximity on P^* (L)^X and the induced (L,M)-fuzzy proximity on P(Λ)^X corresponding to each (L,M)-fuzzy proximity on L^X; L∈L(Λ) are clearly undertaken. Moreover, we have showed that