I was wondering whether the real projective space $\Bbb{R}P^n$ embeds *topologically* into $\Bbb{R}^{n+1}$ for odd $n$.


It certainly doesn't for even $n$ because of Alexander duality. Also it doesn't embed *smoothly* for any $n$. I will prove the two above statements in my algebraic topological class, but I couldn't find anything in the literature for topological embeddings in the odd case. 



Topological embeddings of real projective space in euclidean space
The first page of W.~Massey's paper [On the imbeddability of the real projective spaces in Euclidean space](https://msp.org/pjm/1959/9-3/pjm-v9-n3-p17-p.pdf) states that $\mathbb RP^n$ with $n>1$ cannot be imbedded topologically in $\mathbb R^{n+1}$ because its mod $2$ cohomology algebra does not satisfy a certain condition given by R. Thom. 


The reference points to Theorem V.15 (p.180) of Thom's paper [Espaces fibrés en sphères et carrés de Steenrod,](http://www.numdam.org/article/ASENS_1952_3_69__109_0.pdf)
Annales scientifiques de l'École Normale Supérieure, Série 3 : Volume 69 (1952) , p. 109-182. 


EDIT: Actually, a proof can be found in [Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ for n >1](https://mathoverflow.net/questions/96583/is-it-true-that-all-real-projective-space-rpn-can-not-be-smoothly-embedded-in).


