## Locally Compact Hausdorff Space is Regular

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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 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.

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