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Question concerning compact subtopologies on Hausdorf spaces

  1. Aug 21, 2010 #1

    I'm helping out a student with her upcoming topology exam and something has be stomped. It's probably simple but I'm not seeing it at the moment.

    Consider a Hausdorf space (X,T). Any compact subset of X is therefore closed.

    The question is to prove the existence of a coarser topology on (X,T) so that closed also implies compactness. I'm basically trying to find a coarser topology on X that makes it compact.

    Thanks in advance.
  2. jcsd
  3. Aug 21, 2010 #2


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    What about the trivial topology (= only empty and X are open)?
  4. Aug 21, 2010 #3
    That had occured to me, but I though there might be something you could prove about a topology finer than the trivial one. But I guess it fits the bill.
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