Uniform spaces (Q2702348)

This is mainly a survey-type paper in which the fundamentals of the basic approaches to the theory of uniform structures in the context of \(L\)-fuzzy sets are discussed. The paper consists of an introduction and 5 sections. In the first two sections the author presents a brief survey of the main three approaches to the concept of a uniformity in classical (i.e. crisp) topology. These are the approach based on systems of uniform coverings (J. W. Tukey), the approach based on the so-called ``entourage'' or ``surroundings'' of the diagonal (A. Weil), and a certain modification of the diagonal approach based on special families of functions defined on the exponent of the space \((F:2^X\to 2^X)\). Special accent is made on this last, functional approach, since, on one hand it is not very popular among specialists working in the theory of ordinary (crisp) uniformities, and on the other, this approach was successfully extended to the ``fuzzy case''.NEWLINENEWLINENEWLINEThe last three sections deal with fuzzy uniformities. In the third section the author discusses Hutton \(L\)-fuzzy uniformities -- the concept introduced in 1977 by B. Hutton and later studied by him and other authors. In the present survey the author makes accent to distinguish those definitions and results of this theory which are valid for arbitrary complete lattices \(L\) (and not only for completely distributive ones, as it was assumed in Hutton's papers).NEWLINENEWLINENEWLINEIn the fourth section Lowen fuzzy uniformities are considered -- the concept introduced by Lowen in 1981 and later studied by him and other authors. This approach is based on prefilters in \([0,1]^{X\times X}\) and can be viewed as the generalization of the classical diagonal approach. As different from Hutton's uniformities, the theory of Lowen uniformities heavily relies on the assumption that fuzzy sets take their values in the unit interval \([0,1]\).NEWLINENEWLINENEWLINEFinally, the last, 5th Section presents a new, original, approach to the concept of an \(L\)-fuzzy uniformity. This approach is based on \(L\)-valued filters on the product \(X\times X\) -- i.e. certain mappings of the kind \({\mathcal F}:L^{X\times X}\to L\). As special cases of this general approach U. Höhle's fuzzy uniformities as well as J. Gutierrez García's fuzzy uniformities can be obtained.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00008].