Euclidean neighborhood retract

In mathematics, especially algebraic topology, an Euclidean neighborhood retract or a ENR for short is a topological space that is (or homeomorphic to) a subset of a Eucldean space ${\displaystyle \mathbb {R} ^{n}}$, some n, that is a retract of some neighborhood of the subset.

Definition and results

By definition, a topological space X is called a Euclidean neighborhood retract or an ENR if there is an embedding ${\displaystyle i:X\hookrightarrow \mathbb {R} ^{n}}$ for some n such that ${\displaystyle i(X)}$ is a retract of some neighborhood ${\displaystyle U}$ of it; i.e., there is a map ${\displaystyle r:U\to i(X)}$ such that ${\displaystyle r|_{i(X)}}$ is the identity (such ${\displaystyle r}$ is called a retraction). ^[1] It follows that an ENR is necessarily locally compact and locally contractible in geometric topology sense.

The fundamental result here is the following

Theorem (Borsuk)— ^[2] Let X be a locally compact and locally contractible space. If there is an embedding ${\displaystyle i:X\hookrightarrow \mathbb {R} ^{n},}$ then there is a retraction from some neighborhood U to ${\displaystyle i(X)}$.

The theorem implies in particular that the above retract map r in the definition is actually not part of the data of the definition of an ENR. The theorem also implies many familiar spaces are ENRs; e.g., a smooth manifold, a compact topological manifold, ^[3] a finite CW-complex, ^[4] a real semi-algebraic set are all ENRs. A subset of ${\displaystyle \mathbb {R} ^{n}}$ that is not locally compact, like ${\displaystyle \mathbb {Q} ^{n}}$, is a non-example of an ENR.

See also

• Absolute neighborhood retract

References

1. Bredon 2013 , Appendix E.
2. Bredon 2013, Theorem E.3.
3. Bredon 2013 , Corollary E.5.
4. Bredon 2013 , Corollary E.8.

• Bredon, G.E. (2013). Topology and Geometry. Graduate Texts in Mathematics. Vol. 139. Springer Science & Business Media.
• Hatcher, Allen (2002). Algebraic topology. Cambridge ; New York: Cambridge University Press. ISBN   978-0521795401.

Further reading

• https://math.stackexchange.com/questions/1470120/euclidean-neighbourhoods-retracts-and-deformation-retracts
• https://sites.science.oregonstate.edu/~garity/636/Notes/L02_ANRs2.pdf