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.