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 Euclidean space ${\mathbb{ℝ}}^n$, 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 $i:X\hookrightarrow {\mathbb{ℝ}}^n$ for some n such that $i(X)$ is a retract of some neighborhood $U$ of it; i.e., there is a map $r:U\to i(X)$ such that $r{|}_{i(X)}$ is the identity (such $r$ is called a retraction). 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) —  Let X be a locally compact and locally contractible space. If there is an embedding $i:X\hookrightarrow {\mathbb{ℝ}}^n,$ then there is a retraction from some neighborhood U to $i(X)$.

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 topological manifold, a finite CW-complex, a real semi-algebraic set are all ENRs. A subset of ${\mathbb{ℝ}}^n$ that is not locally compact, like ${\mathbb{\mathbb{Q}}}^n$, is a non-example of an ENR.

See also

• Absolute neighborhood retract

References

• Bredon, G.E. (2013). Topology and Geometry. Graduate Texts in Mathematics. 139. Springer Science & Business Media. 
• Hatcher, Allen (2002). Algebraic topology. Cambridge ; New York: Cambridge University Press. ISBN 978-0521795401. https://pi.math.cornell.edu/~hatcher/AT/ATpage.html. 

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