A new approach towards characterization of semicompactness of fuzzy topological space and its crisp subsets

It is widely accepted that one of the most satisfactory generalization of the concept of compactness to fuzzy topological spaces is $\alpha$-compactness, first introduced by Gantner et al. in 1978, followed by further investigations by many others. Chakraborty et al. introduced fuzzy semicompact set and investigated and characterized fuzzy semicompact spaces in terms of fuzzy nets and fuzzy prefilterbases in 2005. In this paper, we propose to introduce a new approach to characterize the notion of $\alpha$-semicompactness in terms of ordinary nets and filters. This paper deals also with the concept of $\alpha$-semilimit points of crisp subsets of a fuzzy topological space X and the concept of $\alpha$-semiclosed sets in X and these concepts are used to define and characterize $\alpha$-semicompact crisp subsets of X.