Contents

# Contents

## Definition

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

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

## 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)

###### Proposition

A metrisable topological space is an absolute retract precisely if it is

## Examples

###### Example

Every (finite-dimensional) metrizable locally Euclidean topological space – in particular every topological manifold – is an absolute neighbourhood retract.

In fact:

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

###### Proposition

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

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

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

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: