Differential structure on $\mathbb S^n/±\mathrm{id}$ such that the injection $i:\mathbb RP^n\rightarrow\mathbb S^n/(±\mathrm{id}$ is a diffeomorphism

Question

Show that the natural injection $i:\mathbb S^n\rightarrow\mathbb R^{n+1}\backslash\{0\}$ induces a homeomorphism between $\mathbb RP^n$ and $\mathbb S^n/±\mathrm{id}$ equipped with the quotient topology. $\mathbb RP^n$ is the real projective space $\mathbb R^{n+1}/\sim$ where $x\sim y\iff x$ and $y$ are colinear. Show there is a unique differential structure on $\mathbb S^n/±\mathrm{id}$ such that $i$ is a $\mathcal C^\infty$-diffeomorphism.

It is quite clear to understand why $i$ is a homeomorphism since $\mathbb RP^n$ and $\mathbb S^n/±\mathrm{id}$ are the same space but I don't know how to prove it properly. About the second part of the question, It must have something to do with proving that the differential structure looked for is the unique one rendering the projection $\pi:\mathbb S^n\rightarrow\mathbb S^n/±\mathrm{id}$ a local diffeomorphism. I have a feeling that you should equip the space with the atlas of all open sets of $\mathbb S^n$ but I don't really know how to prove it (or even if it's true) and even less how to prove its uniqueness.

PS : I have seen answers for the first part on this website, but they all use a characterization of Hausdorff spaces. Would there be a proof that doesn't use this?

You should edit your question to include the definition of $\mathbb RP^n$. — Paul Frost, Oct 03 '22 at 08:19
Note that the projection $S^n \to S^n/{\pm Id}$ cannot be a diffeomorphism — Didier, Oct 03 '22 at 08:58
I have added the definition. I forgot to write it but it's the structure such that $\pi$ is a $\textbf{local}$ diffeomorphism. — maxjw91, Oct 03 '22 at 11:45

score 1 · Accepted Answer · answered Oct 04 '22 at 00:05

As you say, the the natural injection $i:\mathbb S^n\rightarrow\mathbb R^{n+1}\backslash\{0\}$ induces a continuous map $i': \mathbb S^n/±\mathrm{id} \to \mathbb RP^n$ on the quotient spaces. That $i'$ is a homeomorphism follows from When is the restriction of a quotient map $p : X \to Y$ to a retract of $X$ again a quotient map?
Given a homeomorphism $h : X \to M$ from a space $X$ to a differentiable manifold $M$, there exists a unique differentiable structure on $X$ such that $h$ becomes a diffeomorphism. This is fairly obvious. As a differentiable atlas on $X$ take all $(\phi \circ h, h^{-1}(U))$, where $(\phi, U)$ belongs to the maximal differentiable atlas of $M$.

Differential structure on $\mathbb S^n/±\mathrm{id}$ such that the injection $i:\mathbb RP^n\rightarrow\mathbb S^n/(±\mathrm{id}$ is a diffeomorphism

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