Functor \(\exp ^ c_ n\), absolute retracts and Hilbert space
type: Item
instance of: scholarly article
title: Functor \(\exp ^ c_ n\), absolute retracts and Hilbert space
zbMATH Open document ID: 0611.54010
DOI: 10.1007/BF01157018
author: A. G. Savchenko
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54B30
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 3988226
zbMATH Keywords: Vietoris topology
zbMATH Keywords: absolute (neighborhood) retract of the class of metrizable spaces
zbMATH Keywords: \(\ell _ 1\)-manifold
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01157018
OpenAlex ID: W2015838742
description: scientific article; zbMATH DE number 3988226