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Math Help - Topology

  1. #1
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    Topology

    Let (X,d) be a metric space. X is said to be separable if there is a countable set Q contained in X such that Q is dense in X. Show that a separable metric space has a basis for its open sets consisting of a countable family of open sets.

    I started by establishing the open ball B(q,1/n) where n = 1,2,...

    I know I want to show that every open set is a union of these sets.

    I'm not sure where to go from here. Any help would be great!
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  2. #2
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    You're going to the good direction. Make a drawing in the case of metric space (IRē,||.||).

    I know I want to show that every open set is a union of these sets.
    in fact you need show it just for the open balls because the open balls form a basis of your metric space.
    Last edited by BRom; November 14th 2008 at 11:26 AM.
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  3. #3
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    Let O be an open set. We will prove that every x in O belongs to an open ball B(q_{x},\frac{1}{n_{x}}) with q_{x} \in Q and n_{x} \in \mathbb{N}.
    We know there exists y \in X, r>0 such that  x \in B(y,r), so d(y,x)<r.
    We will note \epsilon=r-d(x,y)

    Let n be an integer such that \frac{1}{n}<\frac{\epsilon}{2}. Since Q is dense in X, there is a q_{x} \in Q\cap B(x,\frac{1}{n}).

    At this point, just visualize the scene, and try to conclude.
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