For any integer $n\ge 1$ the map $q:\mathbb S^n\to\mathbb {RP^n}$, which identifies antipodal points, is a covering map.
I'm trying to solve this question in the following manner (with the help of the comments and answers below):
Let $y$ be a point in $\mathbb {RP^n}$, and take any neighborhood $U$ of $y$. The preimage of $U$ are open subsets $V$ and $-V$, a question emerges, $q|V$ and $q|-V$ are homeomorphic to $U$? and why? if it does so, then we're done?
Am I right?
I'm a beginner in this subject, so I'm not sure if I solved it correctly
Thanks