The alpha-levels of a fuzzy uniform structure and of a fuzzy proximity
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 Gahler et al. (1998) which is defined as some fuzzy filter and we are also interested in the fuzzy proximity N in the sense of Gahler 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 alpha -level uniform structure U-alpha and the alpha -level proximity N-alpha of U and N, respectively. We show that there is one-to-one correspondence between a fuzzy uniform structure U-alpha and the family (U-alpha)(alpha is an element of L0) of uniform structures that fulfills certain conditions, is given by: U alpha = U-alpha and U(U) = boolean OR (A is an element ofU alpha, chiA less than or equal tou) alpha. We also show that the topologies T-U alpha and T-N alpha associated with U-alpha and N-alpha coincides with the alpha -level topologies of the fuzzy topologies tau (U) and tau (N) associated to U and N, respectively, that is, TUalpha = (tau (U))(alpha) and T-N alpha = (tau (N))(alpha). Moreover, we assign for each fuzzy uniform structure U an associated fuzzy proximity of the internal type N-U and hence we get the relation between the alpha -levels of U and of N-U which is given by: N-U alpha = (N-U)alpha. (C) 2000 Elsevier Science B.V. All rights reserved.
fuzzy uniform structures fuzzy proximities fuzzy filters fuzzy topologies homogeneous fuzzy filters stratified fuzzy uniform structures stratified fuzzy proximities stratified fuzzy topologies alpha-level uniform structures alpha-level proximities alpha-level topologies