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  1. #1
    Super Member Aryth's Avatar
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    Topology Question

    Let $\displaystyle \mathbb{R}$ and $\displaystyle \mathbb{R}^2$ have their respective Euclidian topologies. Endow

    $\displaystyle S^1 = \{(x,y) \ \exists \ \mathbb{R}^2 | x^2 + y^2 = 1\}$

    with subspace topology induced from $\displaystyle \mathbb{R}^2$. Define the relation ~ on $\displaystyle \mathbb{R}$ by a ~ b iff $\displaystyle a,b \in \mathbb{Z}$

    Prove that ~ is an equivalence relation on R

    Let [a] denote the ~ equivalence class containing a. Put $\displaystyle \frac{\mathbb{R}}{\mathbb{Z}} := \{[a] | a \in \mathbb{R}\}$ and endow $\displaystyle \frac{\mathbb{R}}{\mathbb{Z}}$ with the quotient topology - that is, the topology induced by the canonical projection

    $\displaystyle n : \mathbb{R} \to \frac{\mathbb{R}}{\mathbb{Z}}$, $\displaystyle a \mapsto [a]$

    Prove that $\displaystyle S^1$ is homeomorphic to $\displaystyle \frac{\mathbb{R}}{\mathbb{Z}}$
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  2. #2
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    The equivalent relation was given wrongly, it should be "$\displaystyle a\sim b$ if and only if $\displaystyle a-b\in \mathbb{Z}$".

    Looking at $\displaystyle \mathbb{R}/\mathbb{Z}$, we see that there is one to one mapping between the classes in $\displaystyle \mathbb{R}/\mathbb{Z}$ and the interval $\displaystyle [0,1)$: for every class $\displaystyle [a]$ there is a unique real number $\displaystyle 0\leq t<1$ such that $\displaystyle t\in [a]$, now map $\displaystyle [a]$ to the unique real number $\displaystyle t$ in $\displaystyle [0,1)$. You can show that it is a homoemorphism.

    We know that $\displaystyle S^1$ can be parametrized by $\displaystyle e^{2\pi it}$ where $\displaystyle 0\leq t<1$, now you see why $\displaystyle S^1\simeq [0,1)$.

    Hope this helps.
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