It seems to be widely known that every topological manifold can be embedded as a neighbourhood retract in euclidean space, I can not find a reference, though.


The reason, why I'm asking this, is that I would like to have a nice class of spaces containing topological manifolds and CW complexes.
And as it is still an open question, whether all 4-manifolds are homeomorphic to a CW complex, I thought I could take something like absolute or euclidean neighbourhood retracts. (In the latter case I do not have all CW complexes, though.)



Every topological manifold is a ENR? (Reference)
As it turns out, a close look at the [Wikipedia page](https://en.wikipedia.org/wiki/Retract#Absolute_neighborhood_retract_.28ANR.29) solves my Question.
There is an (indirect) reference to the paper:


Hanner, Olof (1951), "Some theorems on absolute neighborhood retracts", Arkiv för Matematik, 1: 389–408, [doi:10.1007/BF02591376](http://link.springer.com/article/10.1007%2FBF02591376), MR 0043459.


which proves that being an ANR is a local property. Then it is of course clear that manifolds and locally finite CW complexes are ANR's.
In particular, if we have an embedding of a topological manifold as a closed subset into the Euclidean space, then it is a neighbourhood retract.


The question remains, whether every topological manifold can be embedded as a closed subset into $\mathbb{R}^n$, but this is not that hard. (At least not in the compact case.)


