True. But the mapping is only well defined (independent of representative from the equivalence class) because is periodic over the integers.defines a homemorphism.

is continuous (trivial)Saying it is slightly more subtle, the above is true ONLY because of the construction of . Say more.Letting which implies injective.

And yes it is surjective. But, why do it this way? Really you aren't done, you haven't proven the inverse function is continuous.

Have you proven that is Hausdorff? Then, reverse the direction of your mapping and you only have to show that your function is continuous (you already proved it was bijective) and thus you will have a continuous bijection from a compact space into a Hausdorff space and thus automatically a homeomorphism.

That said, you've already done everything else. Might as well finish it.