How to prove that $S^1 \times S^3$ identifying antipodal points is again $S^1 \times S^3$. I am studying the compactification of Minkowski spacetime as the space of generators $\mathcal{PN}$ of the null cone $\mathcal{N}$ of a six dimensional real manifold of signature 2. Concretely I am following the book "An introduction to twistor theory" by Huggett and Tod
(see the image). Topologically the say that $\mathcal{PN}$ is $S^1\times S^3$ identifying antipodal points (because each generator cuts the 5-sphere twice), which is again $S^1\times S^3$. Does anybody know why $S^1\times S^3/\sim$ is again $S^1\times S^3$?
And can this be extended to $S^1\times S^n$ in general?
Thank you.

$S^1$ and $S^3$ can be identified with the unit spheres in $\mathbb C$ and $\mathbb C^2$ respectively.  Then the map $$(z, w) \mapsto (z^2, z w)$$ where $z \in \mathbb C$, $w \in \mathbb C^2$ and $\lvert z \rvert = \lVert w \rVert = 1$ identifies exactly the antipodal points.
