Covering Space Examples The Projective Plane
Recall from the Covering Spaces page that if $X$ is a topological space then a covering space of $X$ is a pair $(\tilde{X}, p)$ where $\tilde{X}$ is a path connected and locally path connected topological space and $p : \tilde{X} \to X$ is a continuous map such that for every $x \in X$ there exists a path connected open neighbourhood $U$ of $x$ (called an elementary neighbourhood of $x$) such that $p$ restricted to every path component of $p^{-1}(U)$ is a homeomorphism onto $U$.
We will now look at an example of a covering space of $\mathbb{P}^2$.
Let $X = S^2$. Then $S^2$ is a cover of $\mathbb{P}^2$ by choosing a suitable map $p : S^2 \to \mathbb{P}^2$. For every point $p \in \mathbb{P}^2$ we can find a suitably small open neighbourhood of $p$ such that $p^{-1}(U)$ will be two disjoint open neighbourhoods that are antipodal of each other.