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 ${\displaystyle \mathbb {R} ^{n}}$, some n, that is a retract of some neighborhood of the subset.

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

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,^[3][4] a finite CW-complex,^[5] 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.

• Absolute neighborhood retract

• 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.

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