A generalised form of compactness in fuzzy topological spaces

Abstract

The N-compactness of Wang [13], extended in [ 151, provides the first theory of compactness which is applicable to arbitrary fuzzy subsets. Many of the standard properties associated with compactness are valid for N-compactness, including the product theorem. Furthermore N-compactness reduces to ordinary compactness in the topologically generated case. On closer inspection, one finds evidence that a concept weaker than N-compactness might also be desirable. For example, N-compactness does not reduce to any of the standard fuzzified versions of compactness [ 1, 2, 6, 73, in the case where the fuzzy subset under consideration is the whole space. In particular, when applied to the whole space, N-compactness is substantially stronger than fuzzy compactness. As fuzzy compactness seems to be a good working concept for the whole space, an extension of this notion to arbitrary fuzzy subsets might well have applications not accessible to the much stronger N-compactness. As a consequence of the definition, an N-compact fuzzy set is forced to attain a maximum value. As a result, it is possible to have fuzzy sets which are never N-compact, even if the fuzzy topology has only a finite number of open fuzzy sets. Our intuition, based on the case of general topology, would expect every fuzzy set to be “compact” in such a case. In this paper, we introduce a weaker generalisation of compactness, which we call f-compactness. It is shown that ficompactness is inherited by closed fuzzy subsets, continuous images, finite suprema, and products. It reduces to compactness or to fuzzy compactness in all cases where