Fuzzy properties in fuzzy convergence spaces

Abstract

Based on the concept of limit of prefilters and residual implication, several notions in fuzzy topology are fuzzyfied in the sense that, for each notion, the degree to which it is fulfilled is considered. We establish therefore theories of degrees of compactness and relative compactness, of closedness, and of continuity. The resulting theory generalizes the corresponding "crisp" theory in the realm of fuzzy convergence spaces and fuzzy topology.2000 Mathematics Subject Classification: 54A20, 54A40, 54C05, 54D30. Introduction.In most papers and contributions to the theory of [0, 1]-topological spaces, the considered properties (like compactness) are viewed in a crisp way, that is, the properties either hold or fail. In [16], R. Lowen suggested that also the properties should be considered fuzzy, that is, one should be able to measure a degree to which a property holds. There are some papers dealing with such approaches. E. Lowen and R. Lowen , a theory of degrees of precompactness and completeness in the so-called Hutton fuzzy uniform spaces. These latter theories are related to the present work as they are explicitly based on a generalized inclusion (which is, however, not resulting from a residual implication).In this paper, we follow these ideas in a systematic way. Starting from the notion of limit of a prefilter as defined in [15], we consider a semigroup operation * on [0, 1] which is finitely distributive over arbitrary joins and, therefore, has a right adjoint →, that is, a residual implication operator. In this way, a natural way of obtaining truly fuzzy extensions of properties in fuzzy convergence spaces [12,13] is to replace subsethood, a ≤ b of two fuzzy sets a, b. Exploiting this idea leads to the theory considered in this paper. We extend some results of an earlier paper [10], where degrees of closedness and degrees of compactness were studied and a theory of degrees of continuity and of degrees of relative compactness is established. Note that stratified [0, 1]-topological spaces [5,14] as well as Choquet convergence spaces [3] are fuzzy convergence spaces [13], that is, our approach works also in this more special context.