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Locally Compact Hausdorff Space is Regular

  1. Jan 10, 2013 #1
    So, I'm working a bit through munkres and I came across this problem

    Show that every locally compact Hausdorff space is regular.

    So, I think I've solved it, but there is something confusing me. I initially said that if [itex]X[/itex] is locally compact Hausdorff, it has a 1-point compactification, [itex]Y[/itex], which is compact Hausdorff. Then, by some theorem in Munkres (32.3), every compact Hausdorff space is normal. Then again by Munkres, (page 195), a normal space is regular, and a subspace of a regular space is regular, so that [itex]X[/itex] is regular.

    Now, my confusion here is that Munkres defines normal and regular only when 1-point sets are closed. It is not entirely clear to me that this is true for a compact Hausdorff space. Is it? Because if so, then I see that normal implies regular and the proof is done.

    Even without this though, I can see that any compact Hausdorff space is regular purely from the definition of compact and Hausdorff (at least for the definition of regular not using the fact that 1-point sets are closed), and so a subspace of [itex]Y[/itex], namely [itex]X[/itex], must be regular.
     
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  3. Jan 10, 2013 #2

    micromass

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    Singletons are always closed in a Hausdorff space. Try to prove it if you haven't.
     
  4. Jan 11, 2013 #3
    O, that was dumb.

    In general though, do people reguire that normal spaces have 1 point closed sets so that normal implies regularity? It seems that that is what Munkres does.
     
  5. Jan 11, 2013 #4

    micromass

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    It depends on the author. Many authors do not define normal spaces to have closed singletons. Other authors do require it. Munkres seems to require it.
     
  6. Jan 11, 2013 #5
    I wonder then, which of his theorems depend on it, since its not always clear. I think most of his comparisons between regular spaces and normal spaces include the use of a Hausdorff space (well-ordered sets in the order topology, compact Hausdorff sets, metrizable spaces), so it all seems consistent, but I don't know for sure.
     
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