Let p:\mathbb{S}^n \rightarrow \mathbb{R}\mathbb{P}^n (n\geq 1) be the canonical covering projection. For any map f: X \rightarrow \mathbb{R}\mathbb{P}^n define the space

f*X = \{(x,y) \in Xx \mathbb{S}^n | f(x) = p(y) \in \mathbb{R}\mathbb{P}^n\}.

and the covering projection

q: f*X \rightarrow X ; (x,y) \mapsto x

i) What is the fibre of q?

ii) For any space W establish a one-one correspondence between maps g: W \rightarrow f*X and pairs of maps (h:W \rightarrow X, k: W \rightarrow \mathbb{S}^n ) such that fh = pk : W \rightarrow \mathbb{R}\mathbb{P}^n.

iii) Let \mathbb{Z}_2 = \{1, -1\}, the space with two points 1, -1. For f = p :X = \mathbb{S}^n \rightarrow \mathbb{R}\mathbb{P}^n define a homeomorphism e : \mathbb{S}^n x \mathbb{Z}_2 \rightarrow f*X such that

qe : \mathbb{S}^n x \mathbb{Z}_2 \rightarrow \mathbb{S}^n ; (x,y) \mapsto x .

I think i) is relatively straightforward and I get for a given x \in X there are two points a,b \in \mathbb{S}^n with a = -b such that f(x) = p(y) \in \mathbb{R}\mathbb{P}^n. So q^{-1}(x) = \{ (x,a), (x,b) \}.

However I'm struggling with the next two, any help appreciated...