Convergence theory is a primary topic in topology. In fact, topology and so-called Ε½ convergence class are characterized by each other. In fuzzy topology L-fuzzy . topology , more than 40 papers published in the last ten years were concerned with convergence theory. Among these papers, the problem of convergence class was w x w x solved for the case of L s 0, 1 7 . Since the neighbor structure, so called ''quasi-coincident neighborhood system,'' of an L-fuzzy point in an L-fuzzy topological space is in general not directed under the inclusion order, the conditions of w x convergence class in 0, 1 -fuzzy topology will not be valid any longer in the case of Γ 4 Ε½ lattice. Moreover, quite different from the cases of 0, 1 -fuzzy topology i.e., . w x ordinary topology and 0, 1 -fuzzy topology, the so called BolzanoαWeierstrass property does not hold, i.e., a net with a cluster point in an L-fuzzy topological space is not still necessary to have a subnet converging to the point. In this paper, a necessary and sufficient condition for the BolzanoαWeierstrass property is produced, the result is also used in a satisfactory theory of convergence classes in L-fuzzy topological spaces, and the associated characterization theorem between L-fuzzy topologies and convergence classes is established.