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Let $S^n$ be the unit sphere in $\mathbf R^{n+1}$ and $\mathbf R P^n$ be the real projective space(see the definition of $\mathbf R P^n$ I am using in the References).

Define a relation $\sim$ on $S^n$ as: For all $\mathbf v,\mathbf w\in S^n$, $\mathbf v\sim \mathbf w$ if and only if $\mathbf v=-\mathbf w$.

I am trying to show that

$S^n/\sim \ \cong \ \mathbf R P^n$.

Let $\pi:\mathbf R^{n+1}\setminus\{\mathbf 0\}\to \mathbf R P^n$ be the canonical projection and $i:S^n\to \mathbf R^{n+1}\setminus \{\mathbf 0\}$ be the inclusion map.

I want to show that $\pi\circ i:S^n\to \mathbf R P^n$ is a quotient map.

It is clear that $\pi\circ i$ is surjective. We show that $\pi\circ i$ is continuous. Let $U$ be open in $\mathbf R P^n$. Then $\pi^{-1}(U)$ is open in $\mathbf R^{n+1}$. Noting that $i^{-1}(\pi^{-1}(U))=S^n\cap \pi^{-1}(U)$, we infer that $i^{-1}(\pi^{-1}(U))$ is open in $S^n$. Therefore $(\pi\circ i)^{-1}(U)$ is open in $S^n$ and the continuity of $\pi\circ i$ is established.

Now we need to show that if $(\pi\circ i)^{-1}(U)$ is open in $S^n$ for some subset $U$ of $\mathbf R P^n$, then $U$ is open in $\mathbf R P^n$.

I am stcuk here.

Can somebody help?

Thanks.

References:

Define a relation `$\equiv$' on $\mathbf R^{n+1}\setminus \{\mathbf 0\}$ as: for all $\mathbf v,\mathbf w\in \mathbf R^{n+1}\setminus\{\mathbf 0\}$, $\mathbf v\sim \mathbf w$ if and only if $\mathbf v=\lambda \mathbf w$ for some $\lambda\in \mathbf R^\setminus\{0\}$.

Then $\mathbf R P^n=\mathbf R^{n+1}/\equiv$.

caffeinemachine
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1 Answers1

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Since $S^n / {\sim}$ is compact and $\mathbb{R}P^n$ is Hausdorff, it suffices to construct a continuous bijection $S^n / {\sim} \to \mathbb{R}P^n$ - it will be automatically a homeomorphism. But $\pi i : S^n \to \mathbb{R} P^n$ is clearly a continuous surjection where $x,y \in S^n$ have the same image iff $x = \pm y$.

Martin Brandenburg
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