  Given a non-degenerate Peano continuum $X$, a dimension function
$D:2^X_*\to[0,\infty]$ defined on the family $2^X_*$ of compact subsets of $X$,
and a subset $\Gamma\subset[0,\infty)$, we recognize the topological structure
of the system $(2^X,\D_{\le\gamma}(X))_{\alpha\in\Gamma}$, where $2^X$ is the
hyperspace of non-empty compact subsets of $X$ and $D_{\le\gamma}(X)$ is the
subspace of $2^X$, consisting of non-empty compact subsets $K\subset X$ with
$D(K)\le\gamma$.
