Locally Compact Hausdorff Space is Regular

[sammycaps]

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.

 

[micromass]

Singletons are always closed in a Hausdorff space. Try to prove it if you haven't.

 

[sammycaps]

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.

 

[micromass]

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.

 

[sammycaps]

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.