Real projective space cover as a discrete fiber fibration

The quotient is open: for any open $O\subseteq S^n$, its saturation is $q^{-1}q(O)=O\cup (-O)$, open. For $x\in S^n$ let $U_x=\{y:⟨y,x⟩>1\mathrm{/}2\}$. Then $U_x$ and $-U_x$ are disjoint open sets, and $q^{-1}q(U_x)=U_x\bigsqcup (-U_x)$. Each restriction is bijective onto the open $q(U_x)$ and is open by the same saturation argument for open subsets of $U_x$, hence is a homeomorphism. Such sets cover the quotient. Thus $q$ is a two-sheeted covering and is Hurewicz by F1.

Its fiber is discrete with two points. Every based map of a positive-dimensional cube into this fiber is constant, since each line segment in the cube has connected image and a discrete set has no nonconstant path. All positive fiber homotopy groups vanish. F2 then gives $q_{\ast }:{\pi }_k(S^n)\cong {\pi }_k({\mathbb{R}\mathbb{P}}^n)$ for $k\ge 2$, because both adjacent fiber groups are zero, including ${\pi }_1$ of the fiber at $k=2$.

For $n=1$, regard $S^1$ as complex unit numbers. The map $z\mapsto z^2$ is constant on antipodal pairs and has precisely these fibers: $w^2=z^2$ implies $(w-z)(w+z)=0$. It is onto, since $e^{i\theta }$ has square root $e^{i\theta \mathrm{/}2}$. F4 gives a continuous bijection $v:{\mathbb{R}\mathbb{P}}^1\to S^1$. The source is compact as the quotient image of the compact circle by F5–F6, and the target is Hausdorff, so F5 makes $v$ a homeomorphism. The composite $vq$ sends the generator loop $e^{2\pi it}$ to $e^{4\pi it}$, so F7 gives multiplication by two on $\mathbb{Z}$.

For $n\ge 2$, F3 makes the total space path connected and simply connected. The F2 action of ${\pi }_1({\mathbb{R}\mathbb{P}}^n)$ on the two fiber components is transitive because both are in one total-space component, and its stabilizer is the image of ${\pi }_1(S^n)=0$. Its orbit map is therefore a bijection with a two-element set, so the group has exactly two elements. Its nonidentity element squares to identity: its square cannot equal itself by cancellation, leaving only identity. This identifies the group with $\mathbb{Z}\mathrm{/}2$. A path from $x_0$ to $-x_0$ projects to a loop realizing the nontrivial action; its existence follows already from F3.

The spaces and two-point fiber are nonempty. Step 1.3 is the exceptional endpoint $n=1$; it is not covered by the simple-connectivity input in step 2.1. The higher-group computation in step 1.2 remains valid also for $n=1$. There is no claim here for $n=0$, whose total space is disconnected. Every chart was explicitly specified by its centre and all component assignments are unique; no AC or numerable-bundle theorem was used.