The compact Z-set property in convex sets
type: Item
instance of: scholarly article
title: The compact Z-set property in convex sets
zbMATH Open document ID: 0594.52002
DOI: 10.1016/0166-8641(86)90038-6
author: Tadeusz Dobrowolski
published in: Topology and its Applications
publication date: 1986-01-01T00:00:00Z
review text: A metric space X has the compact Z-set property if all compacta of X are Z-sets (a closed subset A of X is a Z-set if any map of any finite- dimensional cell into X is arbitrary closely approximable by a map into \(X\setminus A)\). Let C be an infinite-dimensional, \(\sigma\)-fd-compact (i.e. countable union of finite-dimensional compacta), convex subset of a complete metric linear space. The main result says that C has the compact Z-set property (and hence C is a \(\sigma\)-Z-set in \(\bar C)\). There are given three conditions (e.g., \(\bar C\) not locally compact) each implying that C is homeomorphic to \(\ell^ f_ 2\).
Mathematics Subject Classification ID: 52A07
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 46A55
zbMATH DE Number: 3955873
zbMATH Keywords: convex subsets of metric linear spaces
zbMATH Keywords: homotopically and
zbMATH Keywords: homologically trivial sets
zbMATH Keywords: discrete approximation property
zbMATH Keywords: target-dense embeddings
zbMATH Keywords: compact Z-set property
MaRDI profile type: Publication
full work available at URL: 0166-8641(86)90038-6
OpenAlex ID: W2015731388
cites work: Q4060144
cites work: Characterizing certain incomplete infinite-dimensional absolute retracts
cites work: Discrete cells properties in the boundary set setting
cites work: Boundary sets in the Hilbert cube
cites work: Some Applications of The Topological Characterizations of the Sigma-Compact Spaces l 2 f and &#8721
cites work: Q3903491
cites work: Separable complete ANR's admitting a group structure are Hilbert manifolds
cites work: Q4747245
cites work: Locally Contractible Spaces That are Absolute Neighborhood Retracts
cites work: Characterization of Finite-Dimensional Z-Sets
cites work: Characterizing the Topology of Infinite-Dimensional σ-Compact Manifolds
cites work: Concerning locally homotopy negligible sets and characterization of $l_2$-manifolds
cites work: A wild Cantor set in the Hilbert cube
Recommended article: Characterization of convex \(Z\)-sets in linear metric spaces
Recommended article: Compact convex sets and complex convexity
Recommended article: Q3550723
Recommended article: The Mazur property for compact sets
Recommended article: A geometric characterization concerning compact, convex sets
Recommended article: Q3769518
Recommended article: On the \(P\)-property of convex compact sets.
Recommended article: Compact convex sets with the equal support property
Recommended article: On some criteria of convexity for compact sets
Recommended article: On the Indecomposability of Compact Convex Sets
description: scientific article; zbMATH DE number 3955873