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Prove every Hausdorff topology on a finite set is discret.

  1. Sep 17, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that every Hausdorff topology on a finite set is discrete.
    I'm trying to understand a proof of this, but it's throwing me off--here's why:

    2. Relevant equations
    To be Hausdorff means for any two distinct points, there exists disjoint neighborhoods for those points.
    Also, any finite subset of a Hausdorff space is closed.


    3. The attempt at a solution
    Let a set X have n elements (I'll write it more formal later), but I'll denote them a 1,...,i,...,j,...n.
    For each singleton element, we can write write it as:
    {i} = [itex]\bigcap[/itex](X\{j}) s.t. j≠i.
    And the set {i} is open because it's the intersection of open sets (X\{j}).

    However, isn't that opposite of Hausdorff because both sets are finite subsets.

    Thank you in advance for your help.
     
  2. jcsd
  3. Sep 17, 2012 #2

    Dick

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    {i} is closed, since it's a finite subset of a Hausdorff space. Now you've proved {i} is also open. So it's clopen. Doesn't that make it a discrete space?
     
  4. Sep 17, 2012 #3
    Ah ok!

    My book mentions that every subset of a discrete space is closed, but it doesn't explictly say that it is open when we first discussed them. It mentioned the topology is that every set is open...so I suppose it's implied.

    Thank you!
     
  5. Sep 17, 2012 #4

    Dick

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    Right. If every set S is closed then its complement is also closed. So S is also open.
     
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