I am trying to prove that:
Given $X=R×[-1, 1]$ and the action of $\mathbb Z$ as $m(x,y)=(x+m, (-1)^m y)$, prove that the space $X/Z$ is homeomorphic to the Moebius Band.
Since there is no homeomorphism between $X$ and $[0,1]^2$, where the MB is usually defined, I am trying to work directly on quotients.
My idea is to separate the action of even and odd integers, thus sending part of the orbit of a given $(x,y)$ to $(0,y)$ and the other to $(1, 1-y)$ but I cannot quite formalise it.
I am new both to group action and the forum, so excuse me if my reasoning may bene trivial.