This is the definition of the real projective space in John Lee's book. However what I know is that the real projective space is defined by the quotient space of $S^{n+1}$ by identifying the antipodes. How is the two definitions equivalent? I tried to use the definition of quotient space but cannot find a way to prove the equivalence rigorously... Could anyone please help me?
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You essentially just need to apply the universal property of quotients.
Consider the inclusion map $S^n \hookrightarrow \mathbb{R}^{n+1}$. This is of course continuous, and thus the composition $$\begin{array}{ccccccccc} S^n & \xrightarrow{i} & \mathbb{R}^{n+1} \backslash \{0\} &\\ & \searrow{f} & \downarrow{\pi} \\ & & \mathbb{R}P^n \end{array}$$ is continuous. Note that $f(x)=f(-x)$. Thus, $f$ induces a map $F$ in the quotient by $$\begin{array}{ccccccccc} S^n & \xrightarrow{i} & \mathbb{R}^{n+1}\backslash \{0\} &\\ \downarrow{\pi'} & \searrow{f} & \downarrow{\pi} \\ S^n/\sim & \xrightarrow{F} & \mathbb{R}P^n .\end{array}$$ We have that $F$ is continuous. Note that it is also a bijection. Since $S^n/\sim$ is compact (it is the image by $\pi'$ of $S^n$, which is compact) and $\mathbb{R}P^n$ is Hausdorff (the book may have a proof of that), it is a homeomorphism.
The quotient $\pi: \mathbb{R}^{n+1} \setminus \{0\} \to \mathbb{RP}^n$ can be broken up into two quotients $\mathbb{R}^{n+1} \setminus \{0\} \to S^{n} \to \mathbb{RP}^n$. The first sends $x$ to $\frac{x}{|x|}$ (considering $S^{n}$ as the unit circle embedded in $\mathbb{R}^{n+1}$); the second identifies the antipodes. The composition of these quotients identifies each linear subspace with one point and is precisely $\pi$ as described in that excerpt.