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Math Help - Prove: The intersection of a finite collection of open sets is open in a metric space

  1. #1
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    Red face Prove: The intersection of a finite collection of open sets is open in a metric space

    I want to prove that the intersection of a finite collection of open sets is open in a metric space with metric p.

    Definitions of open and closed:
    A subset M of a metric space X is said to be open if it contains a ball about each of its points. A subset k of X is said to be closed if its complement (in X) is open.

    I found several threads about this subject, but none of those threads offer a profound proof.
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  2. #2
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    Quote Originally Posted by Boyd View Post
    I want to prove that the intersection of a finite collection of open sets is open in a metric space with metric p.
    This is a standard proof by induction.
    First show that if two open sets have a point in common, say x, then there is a ball \mathcal{B}(x;\epsilon) which is a subset of both open sets.
    So the whole proof turns on proving that the intersection of two balls is open.
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