  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 special type of 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.
