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Math Help - separation axioms

  1. #1
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    separation axioms

    prove that :
    every metric space is T3
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  2. #2
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    Quote Originally Posted by flower3 View Post
    prove that :
    every metric space is T3
    What we need to show is that given a closed set C and a point x \not \in C that we can find disjoint open sets around each point.

    define r=\inf \{d(x,y) | y \in C \}

    Now lets make a disjoint open set that covers C

    O = \cup_{y \in C}B_{r/2}(y) since the arbitarty untion of open sets is open O is an open set such that C \subset O

    now consider the ball B_{r/2}(x) is an open set containing x.

    These two sets are disjoint(why?) and fulfill the requirements
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  3. #3
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    These two sets are disjoint(why?)
    basic open sets??!!!
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