How do I give a homeomorphism between $\mathbb R P^n$ and the space obtained by identifying antipodal points of $S^{n+1}$?

Suppose that $Y$ is the quotient space obtained by identifying the antipodal points of $S^ {n+1}$.

I'm trying to give a homeomorphism between $\mathbb R P^n$ and $Y$.

I think that the map $f:\mathbb R^{n+1}-(0)\rightarrow S^n$ taking $x$ to $\frac{x}{||x||}$ will induce a homeomorphism between $\mathbb RP^n$ and $Y$, but I can't make this precise. For example, what exactly will this map be? Any solution will be appreciated.

Note that I'm considering $\mathbb R P^n$ to be the quotient space of $\mathbb R^{n+1}-(0)$ under the equivalence relation $x$~$y$ if $x=ty$ with $t\in\mathbb R-(0)$.

• What is your definition of projective space? – Mankind Jan 20 '15 at 18:35
• You mean $S^n$. – Qiaochu Yuan Jan 20 '15 at 18:41
• Per @user2875124's comment, the answer will depend on the definition of $\mathbb{R}P^n$ you are using. Some people define it as the quotient of $S^n$ under the antipodal map... – Ben Blum-Smith Jan 20 '15 at 18:53

Define the continuous surjection $f:S^n\rightarrow\mathbf{RP}^n$ by $f(x)=[x]$. As this respects the equivalence relation (note $f(\lambda x)=[\lambda x]=[x]=f(x)$), it induces a continuous surjection $S^n/\sim\rightarrow\mathbf{RP}^n$. Note that the latter map is injective; as $S^n/\sim$ is compact and $\mathbf{RP}^n$ is Hausdorff, it follows that this map is even a homeomorphism.