Reading through some lecture notes and it says
The torus $T^2$ is the orientation double cover of the Klein bottle $K$, via the covering projection
$p:T^2\to K; [x,y]\mapsto [x,2y]$
Could someone explain this map? Are they taking $[x,y]$ as the equivalence class of the point $(x,y)$ in $I\times I$ with torus identifications ? I don't see how this maps to the Klein bottle.
Best Answer
Posting so this doesn't go unanswered.