Locally Compact Hausdorff Space is Regular

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Discussion Overview

The discussion revolves around the proof that every locally compact Hausdorff space is regular, as presented in Munkres' text. Participants explore the implications of definitions and theorems related to compact Hausdorff spaces, normal spaces, and regular spaces, while addressing specific conditions regarding closed singletons.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes that a locally compact Hausdorff space has a 1-point compactification, which is compact Hausdorff, leading to the conclusion that it is regular based on Munkres' theorems.
  • Another participant asserts that singletons are always closed in a Hausdorff space, suggesting this supports the regularity of compact Hausdorff spaces.
  • A participant questions whether the requirement for normal spaces to have closed singletons is universally accepted, noting that Munkres appears to impose this condition.
  • There is a suggestion that the dependency of Munkres' theorems on the closed singleton condition is unclear, prompting a discussion about the consistency of his comparisons between regular and normal spaces.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the necessity of closed singletons in the definitions of normal spaces and whether this affects the implications of regularity. There is no consensus on the implications of Munkres' definitions.

Contextual Notes

Participants note that the definitions and theorems discussed may vary by author, leading to potential ambiguities in the application of these concepts. The relationship between normal and regular spaces, particularly in the context of Hausdorff spaces, remains a point of contention.

sammycaps
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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 X is locally compact Hausdorff, it has a 1-point compactification, Y, 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 X 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 Y, namely X, must be regular.
 
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Singletons are always closed in a Hausdorff space. Try to prove it if you haven't.
 
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.
 
sammycaps said:
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.

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.
 
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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