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Hausdorff topology on five-element set that is not the discrete top.

  1. Jan 29, 2013 #1
    1. The problem statement, all variables and given/known data

    The textbook exercise asks for a Hausdorff topology on [itex] \{a,b,c,d,e\}[/itex] which is not the discrete topology (the power set). It is from "Introduction to Topology, Pure and Applied", by Adams and Franzosa.

    2. Relevant equations

    Let X be a set.

    Definition of topology (top for short): must include X and the empty set, must include all intersections of finitely many sets in the top, must include all unions of any sets in the top.

    Definition of Hausdorff: for any two elements x and y of X, there must be disjoint open sets in the top such that one contains x and the other contains y.

    3. The attempt at a solution

    My thought is that this cannot be done. We have a theorem that if X is Hausdorff, then every single point subset of X is closed. This implies that all the four-element subsets of X are (in the topology) and open. But then the intersection of any of the four-element sets generates the rest of the powerset.
     
  2. jcsd
  3. Jan 29, 2013 #2

    micromass

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    That is correct. Any finite Hausdorff space must be discrete!

    Your proof is correct as well!
     
  4. Jan 29, 2013 #3
    Thanks micro :)
     
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