TITLE: Topological embeddings of real projective space in euclidean space
QUESTION [20 upvotes]: 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.

REPLY [14 votes]: The first page of W.~Massey's paper On the imbeddability of the real projective spaces in Euclidean space 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,
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.