Let $mathcal{CL}(X)$ be the space of non-empty closed sets of $X$ with the Vietoris Topology.

In this talk we introduce the concepts of $alpha$-hyperboundedness and pseudo-$omega$-boundedness. We show that, in the subspace of non-empty compact sets $mathcal{K}(X)$ of $mathcal{CL}(X)$, the properties of initially-$alpha$-compactness and $alpha$-boundedness are the same. This result generalizes a known theorem of {rm Du$tilde{s}$an Milovan$tilde{c}$evi'c} from 1985: {it "In $mathcal{K}(X)$ the properties of countable compactness and $omega$-boundedness are the same"}. Also we show that the hyperspace $mathcal{K}(X)$ is pseudocompact if and only if $X$ is pseudo-$omega$-bounded. To finish, we give some results and questions that relates these concepts with the normality of $mathcal{K}(X)$ and the $C^{*}$-embbededness of $mathcal{K}(X)$ in $mathcal{CL}(X)$.