L-fuzzy compactness and related concepts

The compactness defined by Warner and McLean is extended to arbitrary L-fuzzy sets where L is a fuzzy lattice, i.e., a completely distributive lattice with an order reversing involution. It is shown that with our compactness we can build up a satisfactory theory. The different definitions of compactness in L-fuzzy topological spaces are stated and other characterizations of some of these notions are obtained. We also study their goodness and establish the inter-relations between the compactnesses which are good extensions. Good definitions of L-fuzzy regularity and normality are proposed. Following the lines of our compactness we suggest two definitions of L-fuzzy local compactness that are good extensions of the respective ordinary versions. A comparison between them is presented and some of their properties studied. A one point compactification is also obtained. By introducing a new definition of a locally finite family of L-fuzzy sets and combining it with our definition of compactness, we propose an L-fuzzy paracompactness and study some of its properties. Good definitions of L-fuzzy countable and sequential compactness and the Lindelof property are introduced and studied. We also present, in L-fuzzy topological spaces, good extensions of S-closedness and RS-compactness. Some of their properties are examined. Good L-fuzzy versions of almost compactness, near compactness and a strong compactness are put forward and studied. A comparison between these compactness related concepts is also presented