Actually it is one of the exercises of Munkres. $G$ is a group of homeomorphisms of the torus having order $2$. How do I get $G$ in order to make $T/G$ homeomorphic to the Klein bottle? Can anybody give me a hint?
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6If the torus is $S^1\times S^1$, what are obvious self-homeomorphisms of $S^1$ of order $2$? – Thomas Andrews Aug 20 '14 at 18:33
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See also: http://math.stackexchange.com/questions/567304/klein-bottle-covered-by-the-torus?rq=1 and links listed there. – Moishe Kohan Aug 25 '14 at 00:30