# Locally Compact Hausdorff Space is Regular

• sammycaps
In summary, the author has solved a problem but is confused about something. They think that if a locally compact Hausdorff space is regular, then it has a 1-point compactification, Y, which is compact Hausdorff. However, by some theorem in Munkres, every compact Hausdorff space is normal. Therefore, Y must be regular as well.
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.

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.

## 1. What is a locally compact Hausdorff space?

A locally compact Hausdorff space is a topological space that is both locally compact and Hausdorff. This means that for any point in the space, there is a compact neighborhood around it and any two distinct points have disjoint open neighborhoods.

## 2. What does it mean for a locally compact Hausdorff space to be regular?

A locally compact Hausdorff space is regular if for any point and any closed set not containing that point, there exists disjoint open sets containing the point and the closed set respectively. This ensures that the point and the closed set can be separated by disjoint open sets.

## 3. How is regularity different from Hausdorffness?

While Hausdorffness ensures that any two distinct points can be separated by disjoint open sets, regularity extends this to include separating points and closed sets. In other words, regularity is a stronger condition than Hausdorffness.

## 4. Why is regularity important in a locally compact Hausdorff space?

Regularity is important in a locally compact Hausdorff space because it allows for more flexibility in constructing continuous functions and studying the topological properties of the space. Many important theorems and results in topology, such as the Tietze extension theorem and Urysohn's lemma, require regularity.

## 5. Can a locally compact Hausdorff space be regular and not Hausdorff?

No, a locally compact Hausdorff space must be Hausdorff in order to be regular. This is because the Hausdorff property is a necessary condition for regularity. However, there are examples of Hausdorff spaces that are not locally compact and therefore cannot be regular.

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