Geodesic
Whitney blocks in the hyperspace of a finite graph
Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 1, pp. 137-147.

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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 $0$ 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 $C(X)$.
Classification : 05C10, 52B99, 54B20
Keywords: hyperspaces; Whitney levels; Whitney blocks; finite graphs 
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     title = {Whitney blocks in the hyperspace of a finite graph},
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Illanes, Alejandro. Whitney blocks in the hyperspace of a finite graph. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 1, pp. 137-147. http://geodesic.mathdoc.fr/item/CMUC_1995__36_1_a16/