nLab
absolute retract

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

A metrisable topological space YY is an absolute neighborhood retract (ANR) if (Borsuk 32, p. 222) for any embedding YZY \subset Z as a closed subspace in a metrisable topological space ZZ, YY is a neighborhood retract of ZZ.

A metrisable topological space YY is an absolute retract if for any embedding YZY\subset Z as a closed subspace in a metrisable topological space ZZ, YY is a retract of ZZ.

Properties

Proposition

(ANR is a local property for metrizable spaces)
A metrizable topological space which admits an open cover by absolute neighbourhood retracts is itself an absolute neighbourhood retract.

(review in Hu 65, III Thm. 8.1)

(Hu 65, Prop. II.7.2)

Examples

By Prop. (see also Hu 65, III Cor. 8.3).

In fact:

By Palais 1966, Cor. to Thm. 5 on p. 3.

Proposition

Let XX be an absolute neighbourhood retract (ANR) and AiXA \xhookrightarrow{i} X a closed subspace-inclusion. Then AA is an ANR precisely iff the inclusion ii is a Hurewicz cofibration.

(Aguilar, Gitler & Prieto 2002, Thm. 4.2.15)

(Dugundji 52, Kodama 56, review in Hu 65, III Cor. 8.4)

References

The notion of absolute neighbourhood retract is due to

Further development:

  • Olof Hanner, Some theorems on absolute neighbourhood retracts, Arkiv För Matematik Band 1 nr 30 (1950) (doi:10.1007/BF02591376)

  • James Dugundji, Note on CW polytopes, Portugaliae mathematica (1952) 11 1 (1952) 7-10-b (dml:114693)

  • Yukihiro Kodama, Note on an absolute neighborhood extensor for metric spaces, Journal of the Mathematical Society of Japan 8 3 (1956) 206-215 (doi:10.2969/jmsj/00830206)

  • Karol Borsuk, Concerning the classification of topological spaces from the stand point of the theory of retracts, Fundamenta Mathematicae 46 (3) (1959) 321-330 (dml:213516)

Discussion for infinite-dimensional manifolds:

Textbook accounts and review:

See also:

Last revised on September 19, 2021 at 02:30:18. See the history of this page for a list of all contributions to it.