## Relations between the real projective space and canonical covering projections

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

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

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...