Approximate extension property of mappings
type: Item
instance of: scholarly article
title: Approximate extension property of mappings
zbMATH Open document ID: 1295.54024
author: Takahisa Miyata
published in: Topology and its Applications
publication date: 2012-04-13T00:00:00Z
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 6023708
zbMATH Keywords: extension property
zbMATH Keywords: absolute extensor
zbMATH Keywords: extension dimension
zbMATH Keywords: approximate extension property
zbMATH Keywords: absolute approximate extensor
zbMATH Keywords: approximate extension dimension
zbMATH Keywords: shape theory
reviewed by: Vladimir Baladze
MaRDI profile type: Publication
full work available at URL: j.topol.2011.12.011
OpenAlex ID: W1995918054
cites work: A Completely Finitistic Space is Finite-Dimensional
cites work: On certain constructions in finitistic spaces
cites work: Compact Lie group actions on finitistic spaces
cites work: On Intersections of Compacta in Euclidean Space
cites work: EXTENSION OF MAPPINGS INTO $ \mathrm{CW}$-COMPLEXES
cites work: Q3841770
cites work: Cohomological Dimension and Metrizable Spaces
cites work: Extension dimension for paracompact spaces
cites work: On finitistic spaces
cites work: Q3689858
cites work: Finitistic spaces and dimension
cites work: The extension dimension of universal spaces
cites work: Extension dimension of a wide class of spaces
cites work: Q3959112
cites work: On spaces having the weak topology with respect to closed coverings, II
cites work: A Vietoris theorem in shape theory
cites work: A new method in fixed point theory
DOI: 10.1016/J.TOPOL.2011.12.011
review text: The problem of extension of mappings from closed subsets of a given space into nice spaces such as a CW-complex, a polyhedron or an ANR-space over the whole space, is one of the interesting problems of geometric topology. It was studied by many mathematicians (A. Dranishnikov, J. Dydak, I. Ivanšic and L. Rubin and others).NEWLINENEWLINEIn the present paper the author investigates this problem for mappings whose codomains are more general spaces. For this aim he gives a generalization of the extension property and defines the notions of approximate extension property, absolute approximate extensor, approximate extension type, and approximate extension dimension and discusses their well-definedness.NEWLINENEWLINEUsing auxiliary lemmas and propositions obtained in the paper, the author proves that shape equivalent spaces have the same extension type and for each metrizable space \(X\), there exists a minimal element in the class of all approximate extension types of finistic separable metrizable spaces \(Y\) which are absolute approximate extensions of \(X\). Besides, the wedge theorem for absolute approximate extensors is proved, properties of closed mappings whose fibers are absolute approximate extensors are studied and the relation of the approximate extension property to shape retracts is established.
description: scientific article; zbMATH DE number 6023708