Functor \(\exp ^ c_ n\), absolute retracts and Hilbert space
For a metric space X let \(\exp^ c_ nX\) denote the space of all nonempty compact subsets of X having at most n connected components, endowed with the Vietoris topology. The operation \(\exp^ c_ n\) defines a selffunctor of the category of metrizable spaces and continuous maps. The author gives sufficient conditions under which a space of the form \((\exp^ c_ nf)^{-1}\{y\}\) is an absolute (neighborhood) retract of the class of metrizable spaces, or it is an \(\ell_ 1\)-manifold (respectively homeomorphic to \(\ell_ 2)\). The same problem is investigated for spaces of the form \(\exp^ c_ nX\). The provided conditions are in this case not only sufficient, but also necessary. A typical result: \(\exp^ c_ nX\) is an \(\ell_ 2\)-manifold (respectively homeomorphic to \(\ell_ 2)\) iff X is a separable topologically complete (connected) locally connected and nowhere locally compact space.
absolute (neighborhood) retract of the class of metrizable spaces
scientific article; zbMATH DE number 3988226