A comparison of different compactness notions in fuzzy topological spaces

Abstract

In this paper we study the different kinds of compactness notions that have been introduced up to this time. We restrict ourselves to fuzzy topological spaces as defined in [4]. In Section 1 we give some alternative characterizations. For a good definition of fuzzy compactness we will demand that in the special case of topological spaces it coincides with the usual notion of compactness. In Section 2 we show which compactness notions are good extensions and which are not. Moreover, we want the fundamental property of compactness in topological spaces, namely, the Tychonoff theorem on products, to be fulfilled in the more general setting of fuzzy topological spaces. In Section 3 we see for which notions there is a .product theorem. In Section 4 we study the implications that exist between the different notions, and in Section 5 we give some concluding remarks. 1.