The topology of systems of hyperspaces determined by dimension functions

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