Degrees of compactness in -fuzzy convergence spaces and its applications

Based on completely distributive lattices L and M, we define the degrees of compactness of ( L , M ) -fuzzy convergence spaces, ( L , M ) -fuzzy topological spaces, ( L , M ) -fuzzy pseudo-quasi-metric spaces and pointwise ( L , M ) -fuzzy quasi-uniform spaces. It is shown that (1) the Tychonoff Theorem with respect to the compactness degrees holds in ( L , M ) -fuzzy convergence spaces and ( L , M ) -fuzzy topological spaces; (2) the compactness degrees of an ( L , M ) -fuzzy pseudo-quasi-metric space and a pointwise ( L , M ) -fuzzy quasi-uniform space are equal to the compactness degrees of their induced ( L , M ) -fuzzy topological spaces, respectively; (3) an ( L , M ) -fuzzy pseudo-quasi-metric space can induce a pointwise ( L , M ) -fuzzy quasi-uniform space and their compactness degrees are equal.