The topology of systems of hyperspaces determined by dimension functions

Given a non-degenerate Peano continuum $X$, a dimension function $D:2_{\ast }^X\to [0,\mathrm{\infty }]$ defined on the family $2_{\ast }^X$ of compact subsets of $X$, and a subset $\mathrm{\Gamma }\subset [0,\mathrm{\infty })$, we recognize the topological structure of the system $(2^X,{\text{D}}_{\le \gamma }(X){)}_{\alpha \in \mathrm{\Gamma }}$, 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 $D(K)\le \gamma$.