LG-paracompactness of LG-fuzzy topological metric spaces

In this manuscript, we introduce LGc -fuzzy Euclidean topological space in which L denotes a completely distributive lattice with a countable subset dense in it. We use the structure of LG-fuzzy topological space (X, T), which X is an L-fuzzy subset of the crisp set M and T : LM X → L, is an L-gradation of openness on X to define the fundamental concepts of LG-fuzzy analysis such as LG-locally compactness and LG-paracompactness and prove several theorems. In consequence, we show that any second countable Hausdorff LG-fuzzy topological space that is LG-locally compact is LG-paracompact. Also from any given metric ρ on a crisp set M and L-fuzzy subset X of it, we construct an Lgradation of openness Tρ on X and obtain LG-fuzzy topological metric space (X, Tρ). Finally, we prove an interesting theorem: Every LG-fuzzy topological metric space, is LG-paracompact.