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Abstract. In this paper we introduce and study the concept of fuzzifying separation axioms in fuzzifying generalized topological spaces.

Compactness and properties closely related to compactness play an important role in the applications of General Topology to Real Analysis and Functional Analysis. In the framework of topological spaces several modified forms of compact spaces have been introduced and studied: nearly compact spaces, mildly compact spaces etc. In this paper, we give some new characterizations of b-compact sets and bclosed sets by means of nets and filterbases. PreliminariesThroughout this paper, spaces always means topological spaces on which no separation axioms are assumed unless otherwise mentioned and b-open, b-closed) sets of (X, τ ) is denoted by BR(X) (resp. BO(X), BC(X)). The family of all b-regular (resp. b-open, b-closed) sets of (X, τ ) containing a point x ∈ X is denoted by BR(X, x) (resp. BO(X, x), BC(X, x)). We will give several characterizations of the b-compact spaces. The first characterization makes use of the finite intersection condition.