A new fuzzy compactness defined by fuzzy nets

Abstract

Compactness is one of the most important notions in topology and is thoroughly investigated in general topology. But it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology. In fact, it would seem that there was not even a good definition of compactness in this area until recently. Indeed, soon after the famous article written by Zadeh [8] was published, Chang [ 1 ] introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given. From then on, a series of different notions of fuzzy compactness have been discussed [2,3, 71, and in [5] they were compared in detail. But it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset; therefore they are far from the crisp compactness. For example, it is impossible to discuss the hereditary property (with respect to closed subsets) of fuzzy compactness. In 1977, Pu and Liu introduced the notion of fuzzy nets and a new concept of so-called Q-neighborhood was given, which could reflect the features of neighborhood structure in fuzzy topological spaces, and by this new neighborhood structure the Moore-Smith convergence theory was established spendidly [6]. It furnishes us a basic tool to investigate the fuzzy compactness in a more general way; to investigate the compactness of arbitrary fuzzy subsets. In this paper a new fuzzy compactness defined by fuzzy nets, we shall call it N-compactness, is given. But we introduce the notion of R-neighborhood (see Section 2) instead of Qneighborhood to establish the Moore-Smith convergence theory, for it is perhaps more convennient to use. N-compactness possesses the following properties: (1) N-compactness is defined for arbitrary fuzzy subsets. (2) For N-compactness the hereditary property with respect to closed subsets holds. (3) N-compactness is a good extension [5]. (4) The continuous images of Ncompact sets are N-compact. (5) For N-compactness the Tychonoff product theorem holds. (6) If an N-compact fuzzy topological space is T2 [6], then it will satisfy a stronger separation axiom which will be called T4 in this paper. 1 0022-247X/83 $3.00