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Thread: homeomorphism from a partition of R^2 to (S^1) x (S^1)

  1. #1
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    homeomorphism from a partition of R^2 to (S^1) x (S^1)

    Quote Originally Posted by Problem Statement
    Find an equivalence relation $\displaystyle \sim$ on $\displaystyle \mathbb{R}\times\mathbb{R}$ such that $\displaystyle \mathbb{R}\times\mathbb{R}/\sim$ is homeomorphic to $\displaystyle S^1\times S^1$.
    (Recall that $\displaystyle S_1:=\{x\in\mathbb{R}^2:|x|=1\}$ is the unit circle in $\displaystyle \mathbb{R}^2$.)

    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 $\displaystyle \sim$ as the relation induced by the partition $\displaystyle P\times P$ of $\displaystyle \mathbb{R}^2$, where

    $\displaystyle P:=\{(-\infty,0]\cup[2\pi,\infty),\{\theta\}:\theta\in(0,2\pi)\}$.

    We define the bijection $\displaystyle \varphi$ from $\displaystyle S_1$ onto $\displaystyle P$ by $\displaystyle \varphi(0)=(-\infty,0]\cup[2\pi,\infty)$ and $\displaystyle \varphi(\theta)=\{\theta\}$ whenever $\displaystyle \theta\in(0,2\pi)$. Then the map from $\displaystyle S_1\times S_1$ to $\displaystyle P\times P=\mathbb{R}^2/\sim$ defined by $\displaystyle (\theta_1,\theta_2)\mapsto(\varphi(\theta_1),\varp hi(\theta_2))$ 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?

    Thanks!
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  2. #2
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    From "http://en.wikipedia.org/wiki/Torus#Topology",
    "The torus can also be described as a quotient of the Cartesian plane under the identifications

    (x,y) ~ (x+1,y) ~ (x,y+1).
    "
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