# Reference question: space of embeddings

Let $X$ be a compact surface of genus $g \geq 1$. Then it is a well known fact that the space of embeddings into $\mathbb{R}^{\infty}$ is contractible. The proof uses Whitney's embedding theorem. Moreover, this space a CW-complex by simplicial approximation on embedding spaces $X \rightarrow \mathbb{R}^n$.

Is there an article or text book that I may refer to. All papers I read just mention it without proof.

-
There is a proof (without using Whitney's embedding theorem) and some references at nlab.mathforge.org/nlab/show/embedding+of+smooth+manifolds –  Loop Space Mar 26 '13 at 10:45
Thank you for this link. Do you know some other reference where the proof is written down explicitly? –  berl13 Mar 26 '13 at 10:57

It is for $\ell^2$ there also with a universal Ehresmann connection. The proof adapts to $\mathbb R^{(\infty)}$ as used in section 47 of (here).