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Homework Help: Point in a topological space

  1. Sep 24, 2007 #1
    1. The problem statement, all variables and given/known data
    Is it true that every point in a topological space is closed? In a metric space?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 24, 2007 #2


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    Have you tried constructing a counterexample?

    A single point space clearly won't suffice; what about a two-point space?
  4. Sep 24, 2007 #3


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    Try looking up T1 spaces.
    Then look at Hausdorff spaces and metric spaces and try to prove whether they are in general T1.
  5. Sep 24, 2007 #4
    you can also think about discrete topology, if you have learned before.
  6. Sep 24, 2007 #5
    So, I found it is only true in a T1 space.

    But in a single point space it is also true since the complement of every point is the null set which is open.
  7. Sep 24, 2007 #6


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    Actually, it might be better to think about the indiscreet topology rather than the discreet topology.
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