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Thread: Distance function: continuous?

  1. April 5th 2009, 03:38 PM #1
    Andreamet
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    Distance function: continuous?

    Let (X,d) be a metric space. Prove that the distance function
    d: X \times X \rightarrow \mathbb{R} is continuous, assuming that XxX has the product topology that results from each copy of X having the topology induced by d.
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  2. April 5th 2009, 04:40 PM #2
    Mush
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    Quote Originally Posted by Andreamet View Post
    Let (X,d) be a metric space. Prove that the distance function
    d: X \times X \rightarrow \mathbb{R} is continuous, assuming that XxX has the product topology that results from each copy of X having the topology induced by d.
    Remember, a function f:R \rightarrow R is said to be continuous if, at the point a \in R, if given \epsilon > 0 , there is a \delta > 0 such that

    |f(x) - f(a)| < \epsilon

    whenever:

    |x-a| < \delta
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