RP^n Projective n-space

Define an equivalence relation on S^n\subset\mathbb{R}^{n+1} by writing v\sim w if and only if v=\pm w. The quotient space P^n=S^n/\sim is called projective n-space. (This is one of the ways that we defined the projective plane P^2.) The canonical projection \pi: S^n\to P^n is just \pi(v)=\{\pm v\}. Define U_i\subset P^n, 1\leq i\leq n+1, by setting \displaystyle U_i=\{\pi(x^1, \dots, x^{n+1})\mid x^i\neq 0\}.

Prove
1) U_i is open in P^n.
2) \{U_1, \dots, U_{n+1}\} covers P^n.
3) There is a homeomorphism \varphi_i: U_i\to\mathbb{R}^n.
4) P^n is compact, connected, and Hausdorff, hence is an n-manifold.

Proof:
1) \pi^{-1}U_i=\{(x^1, \dots, x^{n+1})\mid x^i\neq 0\} is open in S^n, so U_i is open in P^n.
2) Let y=\pi(x^1,\dots, x^{n+1})\in P^n. Then since (x^1,\dots, x^{n+1})\neq(0,\dots,0), so y\in\bigcup_{i=1}^{n+1}U_i. Hence P^n\subset\bigcup_{i=1}^{n+1}U_i.
3) Consider A=\{(x^1,\dots, x^{n+1})\mid x^i+1\}\cong\mathbb{R}^n. Define \displaystyle \varphi_i(\pi(x^1,\dots, x^{n+1}))=(\frac{x^1}{\|x^i\|},\dots,\frac{x^{i-1}}{\|x^i\|},1,\dots,\frac{x^{n+1}}{\|x^i\|}) for x^i>0. If x^i<0, then \varphi_i(\pi(x^1,\dots, x^{n+1}))=\varphi_i(\pi(-x^1,\dots, -x^{n+1})). Then \varphi_i is well-defined.

\displaystyle \varphi_i^{-1}(x^1,\dots,1,\dots,x^{n+1})=\pi(\frac{x^1}{\|v\|},\dots,\frac{1}{\|v\|},\dots,\frac{x^{n+1}}{\|v\|}), where v=(x^1,\dots, 1,\dots, x^{n+1}). Both \varphi_i and \varphi_i^{-1} are continuous, so \varphi_i: U_i\to A is a homeomorphism.
4) Since S^n is compact and connected, so is P^n=S^n/\sim. P^n is a CW-complex with one cell in each dimension, i.e.\ P^n=\bigcup_{i=0}^n e^n. Since CW-complexes are Hausdorff, so is P^n.

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