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Compact -> hausdorff

  1. Sep 6, 2009 #1
    Is there a way to make a compact space hausdorff while preserving compactness?
     
  2. jcsd
  3. Sep 6, 2009 #2
    Specify what you mean.
    A sphere is compact and Hausdorff.
     
  4. Sep 6, 2009 #3
    I am not looking for an example of a compact & hausdorff space.

    I am asking -- if I have a compact space, is there a process by which I can make it a hausdorff space while preserving compactness.

    In a sense, I am looking for something like one point compactification(but not that, more like "hausdorffication").
     
  5. Sep 6, 2009 #4
    Just a thought... if the border of your compact set is a manifold, I think you can usually find a continuous mapping that extends the set into itself when going over the border. But since for example finite sets are compact, I don't see a sane way of making every compact set Hausdorff.
     
  6. Sep 6, 2009 #5

    Hurkyl

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    What are you looking to preserve? What is the application?

    There are certainly things you can do: e.g. you can identify inseparable points. This works well for something like a Euclidean line with a double point at the origin (which gives you the Euclidean line). This doesn't work well for something like the Zariski plane over a field. (the result is the one-point space).
     
  7. Sep 6, 2009 #6
    That statement sounds stupid in retrospective sorry...
     
  8. Sep 6, 2009 #7
    Perhaps he means this: Given a compact space X (not Hauseorff), can you enlarge the topology to make it a compact Hausdorff space? If that is what he means, then the answer is, in general, "no".
     
  9. Dec 22, 2009 #8
    Here is a way:

    Take any topological space. Remove all points but one. This space is Hausdorff and compact.
     
  10. Dec 22, 2009 #9
    Ok, good point.

    I should clarify that originally I was asking for some embedding of a compact space which turns out to be hausdorff.
     
  11. Dec 22, 2009 #10
    This is a topological property. You can't alter either of those without altering the topology. Do you mean: is there a Hausdorff space which admits a compact non-Hausdorff subspace? The answer to that is no.
     
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