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I'm self-studying algebraic topology and need help with the following problem (I'm only at part a.)

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The relevant definitions are as follows.

Definition: Let F be a discrete space and X be any space. Then X $\times$ F is a disjoint union of copies of X, indexed by F. The projection $\pi:$ X $\times$ F $\to$ X is called a trivial covering.

Definition: A map $p:\tilde{X} \to X$ is a covering map if it is locally a trivial covering. That is, if X has an open cover $\{N_\alpha: \alpha \in A\}$ by trivializing neighborhoods for p, i.e. there exists discrete $F_\alpha$, and a homeomorphism $\varphi_\alpha: p^{-1}(N_\alpha) \to N_\alpha \times F_\alpha$ such that $p = \pi \circ \varphi_\alpha$ on $p^{-1}(N_\alpha)$.

In terms of diagram, my interpretation of the problem is as follows.

enter image description here

To show that $p$ is a covering map, I guess I'd have to find the discrete space $F$ and the homeomorphism $\varphi_\alpha$, but I don't know how to proceed. Also, I understand that $\mathbb{R}P^n$ is the space obtained by identifying antipodal points of $S^n$, but I can't figure out what its open sets look like.

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Your picture is misleading because it assumes that $p$ is a trivial covering. This is not true, it is only a locally trivial covering. To see that, define $U_i^\pm = \{ (x_1,\dots,x_{n+1}) \in S^n \mid (-1)^{\pm 1} x_i > 0\}$. These set are the intersections of $S^n$ with open half-spaces in $\mathbb{R}^{n+1}$, thus open subsets of $S^n$. Note that the $U_i^\pm$ cover $S^n$.

We have $x= (x_1,\dots,x_{n+1}) \in U_i^+$ if and only if $- x = (-x_1,\dots,-x_{n+1}) \in U_i^-$. Hence $p(U_i^+) = p(U_i^-)$, and we denote this subset of $\mathbb{R}P^n$ by $V_i$.

$V_i$ is open in $\mathbb{R}P^n$ because $p^{-1}(V_i) = U_i^+ \cup U_i^-$. Clearly $p_i^\pm : U_i^\pm \stackrel{p}{\rightarrow} V_i$ is a bijection. It is even a homeomorphism because it maps open sets to open sets.

Now let $F = \{+1,-1 \}$. Then we get a homeomorphism $$\phi_i : p^{-1}(V_i) \to V_i \times F, \phi_(x) = \begin{cases} (p_i^+(x),+1) & x \in U_i^+ \\ (p_i^-(x),-1) & x \in U_i^- \end{cases} $$

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