The Mobius band:

Consider the strip $Y = [0, 1]\times \mathbb{R}$ and let $S:Y\longrightarrow Y$ be the homeomorphism

$$S(x, y) = (1-x, y+1)$$

of $Y$. Then $S$ generates an infinite cyclic group of homeomorphisms of $Y$ acting properly discontinuously on $Y$. The resulting orbit space is the Möbius band. It is an exercise to show that the cylinder is a double cover of the Möbius band.