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.

By Prop. . (review in Hu 65, III Cor. 8.3)

###### Example

(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)

• J. 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

• 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)

Textbook accounts and review: