# Prove that quotient map is covering map

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

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 $$\phi_i : p^{-1}(V_i) \to V_i \times F, \phi_(x) = \begin{cases} (p_i^+(x),+1) & x \in U_i^+ \\ (p_i^-(x),-1) & x \in U_i^- \end{cases}$$