In this paper, we present the characterizations of total boundedness, relative compactness and compactness in fuzzy set spaces equipped with the endograph metric. The conclusions in this paper significantly improve the corresponding conclusions given in our previous paper [H. Huang, Characterizations of endograph metric and \(\Gamma \) -convergence on fuzzy sets, Fuzzy Sets and Systems 350 (2018), 55–84]. The results in this paper are applicable to fuzzy sets in a general metric space. The results in our previous paper are applicable to fuzzy sets in the m-dimensional Euclidean space \(\mathbb {R}^m\) , which is a specific metric space. Furthermore, based on the above results, we give the characterizations of relative compactness, total boundedness and compactness in a kind of common subspaces of general fuzzy sets according to the endograph metric. As an application, we investigate some relationship between the endograph metric and the \(\Gamma \) -convergence on fuzzy sets.
