  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 fist 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 compact
positive $\al$-cuts fuzzy sets 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.
