Degrees of compactness in (L,M)(L,M)-fuzzy convergence spaces and its applications

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