TITLE: Torus as double cover of the Klein bottle
QUESTION [14 upvotes]: 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.

REPLY [8 votes]: Posting so this doesn't go unanswered.

If you know the way of identifying opposite edges of a square to make
  a Klein bottle or a torus, take a 2x1 rectangle and identify edges to
  make a torus. Draw a line down the middle of the rectangle dividing it
  into two squares and with the middle line having opposite orientation
  to the parallel sides. If you identify both ends with the centre, this
  does not disturb the identification of the ends with each other
  (relative orientation is preserved), each square makes a Klein bottle
  - and you can see your double cover. –  Mark Bennet May 3 '12 at 15:51