  This paper discusses the properties the spaces of fuzzy sets in a metric
space equipped with the endograph metric and the sendograph metric,
respectively. We first give some relations among the endograph metric, the
sendograph metric and the $\Gamma$-convergence, and then investigate the level
characterizations of the endograph metric and the $\Gamma$-convergence. By
using the above results, we give some relations among the endograph metric, the
sendograph metric, the supremum metric and the $d_p^*$ metric, $p\geq 1$. On
the basis of the above results, we present the characterizations of total
boundedness, relative compactness and compactness in the space of fuzzy sets
whose $\alpha$-cuts are compact when $\alpha>0$ equipped with the endograph
metric, and in the space of compact support fuzzy sets equipped with the
sendograph metric, respectively. Furthermore, we give completions of these
metric spaces, respectively.
