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Math Help - Topology Question

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

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

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

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

    Prove that ~ is an equivalence relation on R

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

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

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

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

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

    Hope this helps.
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