Degrees of compactness in fuzzy convergence spaces
In [ibid. 92, No. 3, 349-355 (1997; Zbl 0968.54502)] the author introduced a notion of compactness for fuzzy subsets of fuzzy convergence spaces in the sense of [\textit{E. Lowen}, \textit{R. Lowen}, and \textit{P. Wuyts}, ibid. 40, No. 2, 347-373 (1991; Zbl 0728.54001)]. The aim of this paper is to define the measure of compactness of a fuzzy set in a fuzzy convergence space -- that is to assign to a fuzzy set a certain number \(\alpha \in [0,1]\) characterizing ``to what degree this fuzzy set is compact''. The definition of the degree of compactness relies on the concept of residual implication which allows ``to fuzzify'' the crisp subsethood relation. The theory which is being developed in the paper generalizes at the same time the theory of compactness for fuzzy convergence spaces worked out by the author earlier and some measures of compactness in (Chang) fuzzy topological spaces introduced by Lowen. Among others, the author is interested in applying the notion of degree of compactness for the study of function spaces in fuzzy topology.
scientific article; zbMATH DE number 1743120