Let $𝒟$ denote a true dimension function, i.e., a dimension function such that $𝒟\left({ℝ}^n\right)=n$ for all $n$. For a space $X$, we denote the hyperspace consisting of all compact connected, non-empty subsets by $C\left(X\right)$. If $X$ is a countable infinite product of non-degenerate Peano continua, then the sequence ${\left({𝒟}_{\ge n}\left(C\left(X\right)\right)\right)}_{n=2}^{\infty }$ is $F_{\sigma }$-absorbing in $C\left(X\right)$. As a consequence, there is a homeomorphism $h:C\left(X\right)\to Q^{\infty }$ such that for all $n$, $h\left[\{A\in C\left(X\right):𝒟\left(A\right)\ge n+1\}\right]=B^n\times Q\times Q\times \cdots$, where $B$ denotes the pseudo boundary of the Hilbert cube $Q$. It follows that if $X$ is a countable infinite product of non-degenerate...

Let $X$ be a finite graph. Let $C\left(X\right)$ be the hyperspace of all nonempty subcontinua of $X$ and let $\mu :C\left(X\right)\to ℝ$ be a Whitney map. We prove that there exist numbers $0<T_0<T_1<T_2<\cdots <T_M=\mu \left(X\right)$ such that if $T\in (T_{i-1},T_i)$, then the Whitney block ${\mu }^{-1}(T_{i-1},T_i)$ is homeomorphic to the product ${\mu }^{-1}\left(T\right)\times (T_{i-1},T_i)$. We also show that there exists only a finite number of topologically different Whitney levels for $C\left(X\right)$.

The paper solves one problem by E. Prisner concerning the 2-distance operator T₂. This is an operator on the class $C_f$ of all finite undirected graphs. If G is a graph from $C_f$, then T₂(G) is the graph with the same vertex set as G in which two vertices are adjacent if and only if their distance in G is 2. E. Prisner asks whether the periodicity ≥ 3 is possible for T₂. In this paper an affirmative answer is given. A result concerning the periodicity 2 is added.