$\operatorname{Hom}_{\mathbb{Z}_2}(S^n,\mathbb{R}) \simeq S^n \times_{\mathbb{Z}_2} \mathbb{R}$

Question

Let $\operatorname{Hom}_{\mathbb{Z}_2}(S^n,\mathbb{R})$ the set of $\mathbb{Z}_2$-maps from $S^n$ to $\mathbb{R}$ and $S^n \times_{\mathbb{Z}_2} \mathbb{R}$ the fiber product of $S^n$ and $\mathbb{R}$. How can I prove that that $ \operatorname{Hom}_{\mathbb{Z}_2}(S^n,\mathbb{R}) = S^n \times_{\mathbb{Z}_2} \mathbb{R} $? More generally, how can I prove that there is a 1-to-1 corrispondence among $\operatorname{Hom}_{\mathbb{Z}_2}(S^n,\mathbb{R})$ and the set of the cross sections $\Gamma(S^n \times_{\mathbb{Z}_2} \mathbb{R} \to \mathbb{R}P^n)$?