Admissible fibers and \(T\)-admissibility in hyperspaces
Let \(X\) be a metric continuum. Denote by \(2^X\) and \(C(X)\) the hyperspaces of all nonempty closed subsets and all nonempty subcontinua of \(X\), respectively. The space \(X\) is said to be \(T\)-admissible if it admits a Whitney map \(\mu : C(X) \to [0, \mu(X)]\) having some special properties that are too technical to repeat in this review. It is known that hyperspace contractibility for \(X\) implies \(T\)-admissibility for \(X\). In this interesting paper the author proves among other things that \(X\) is \(T\)-admissible if and only if \(C(X)\) is \(T\)-admissible. He also presents an example of a \(T\)-admissible continuum \(X\) such that \(C(X)\) is not contractible. As a consequence, \(C(X)\) is \(T\)-admissible and therefore does not contain any \(R^i\)-continuum. This answers a question of Charatonik.
scientific article; zbMATH DE number 792532
scientific article