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hyperspaces; Whitney levels; Whitney blocks; finite graphs
Let $X$ be a finite graph. Let $C(X)$ be the hyperspace of all nonempty subcontinua of $X$ and let $\mu :C(X)\rightarrow \Bbb R$ be a Whitney map. We prove that there exist numbers $0<T_0<T_1<T_2<\dots <T_M=\mu (X)$ 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 $\mu ^{-1}(T)\times (T_{i-1},T_i)$. We also show that there exists only a finite number of topologically different Whitney levels for $C(X)$.
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