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Homework Help: Metric Space, Show that it's open

  1. Feb 7, 2010 #1
    Let (X,d) be a metric space, and x is an element in X. Show that [tex]\{y \in X|d(y,x)>r\}[/tex] is open for all r in Reals.

    I really need some help with this one, I have almost no idea on how I am meant to solve this.

    The only thing i know is that I have to use the Openness definition, that states something like [tex]\forall x_0 \in U \exists r>0| B_r \in U[/tex], where in U is a subelement of the metric space (X,d).

    But i don't know how to get started.
     
  2. jcsd
  3. Feb 7, 2010 #2

    HallsofIvy

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    Let a be a point in {y| d(x,y)> r}. Then d(x,a)> r. Construct the neighborhood about a with radius (d(x,a)- r)/2. If b is any point in that neighborhood, use the triangle inequality to show that d(x, b)> r also.
     
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