"The torus can also be described as a quotient of the Cartesian plane under the identifications
(x,y) ~ (x+1,y) ~ (x,y+1).
(Recall that is the unit circle in .)Originally Posted by Problem Statement
This is my first time dealing with quotient topologies and unit spheres and such, so I'm a little unsure. I was thinking of defining as the relation induced by the partition of , where
We define the bijection from onto by and whenever . Then the map from to defined by is a homeomorphism... or at least I think it is.
My first question is, do I really have a solution, or have I made a mistake? My second question is, assuming that my solution works, how do I prove it?