The α-levels of a fuzzy uniform structure and of a fuzzy proximity

Abstract

There are different notions of fuzzy uniform structures and of fuzzy proximities that have been introduced in the literature. In this paper we are interested in the fuzzy uniform structure U in the sense of Gähler et al. (1998) which is defined as some fuzzy filter and we are also interested in the fuzzy proximity N in the sense of Gähler et al. (1998), called the fuzzy proximity of the internal type that is defined by means of another notion of symmetry not depending on an order-reversing involution. Here, we introduce the α-level uniform structure U α and the α-level proximity N α U and N , respectively. Weshow that there is one-to-one correspondence between a fuzzyuniform structure U and the family ( U α ) α∈ L 0 of uniform structuresthat fulfills certain conditions, is given by: U α= U α and U(U)=⋁ A∈U α,χA⩽u α . We also show that the topologies T u α and T N α associated with U α and N α coincides with the α-level topologies of the fuzzy topologies τ U and τ N associated to U and N ,respectively, that is, T U α =(τ U ) α and T N α =(τ N ) α . Moreover, we assign for each fuzzyuniform structure U an associated fuzzy proximity of theinternal type N U and hence we get the relationbetween the α-levels of U and of N U which is given by: N U α =( N U ) α .