Many applications of fuzzy sets restrict attention to the convenient metric space (

${\mathcal{E}}^{n},D)$ of normal, fuzzy convex sets on the base space

${\mathbb{R}}^{n}$, with D the supremum over the Hausdorff distances between corresponding level sets. We mention in particular the fuzzy random variables of

*M. L. Puri* and

*D. A. Ralescu* [Ann. Probab. 13, 1373-1379 (1985;

Zbl 0583.60011)], the fuzzy differential equations of

*O. Kaleva* [Fuzzy Sets Syst. 24, 301-317 (1987;

Zbl 0646.34019)], the fuzzy dynamical systems of the second author [Fuzzy Sets Syst. 7, 275-296 (1982;

Zbl 0509.54040)] and the chaotic iterations of fuzzy sets of Diamond and Kloeden. In these papers specific results are often obtained for compact subsets of

${\mathcal{E}}^{n}$, which raises the question of how to characterize such compact subsets. The purpose of this is to present a convenient characterization of compact subsets of the metric space (

${\mathcal{E}}^{n},D)$. Our main result is that a closed subset of

${\mathcal{E}}^{n}$ is compact if and only if the support sets are uniformly bounded in

${\mathbb{R}}^{n}$ and the support functions of Puri and Ralescu are equileftcontinuous in the membership grade variable

$\alpha $ uniformly on the unit sphere

${S}^{n-1}$ of

${\mathbb{R}}^{n}$. To this end we note that the support functions provides a means of embedding all of the space

${\mathcal{E}}^{n}$ in a Banach space, which we exhibit explicitly, not just the subspace

${\mathcal{E}}_{Lip}^{n}$ of ‘Lipschitzian’ fuzzy sets considered by Puri and Ralescu.