Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In proving that a locally compact Hausdorff space $X$ is regular, we can consider the one-point compactification $X_\infty$ (this is not necessary, see the answer here, but bear with me). Since $X$ is locally compact Hausdorff, $X_\infty$ is compact Hausdorff. As a result, $X_\infty$ is normal.

Imitating the idea in my proof in the link above and taking into consideration the correction pointed out in the answer, let $A,B\subseteq X$ be two disjoint, closed sets in $X$, and consider $X_\infty$. Since $X_\infty$ is normal, there are disjoint open sets $U,V \subseteq X_\infty$ such that $A\subseteq U$ and $B \subseteq V$. Then considering $X$ as a subset of $X_\infty$, we can take the open sets $U \cap X$ and $V \cap X$ as disjoint, open subsets of $X$ that contain $A$ and $B$, respectively...or so I thought. I see from the answers to this question that this does not succeed in proving that a locally compact Hausdorff space is normal, since this is not true.

So my question is simply: why does the above proof fail?


share|cite|improve this question
up vote 6 down vote accepted

$A$ and $B$ are closed in $X$. They need not be closed in the compactification $X_\infty$. You could try to fix this by replacing them with their closures in $X_\infty$, but then these need not be disjoint.

share|cite|improve this answer
A specific example: The Tikhonov plank is the one-point compactification of the deleted Tikhonov plank $X$, which is not normal: the sets $H$ and $K$ of this proof of non-normality are closed and disjoint in $X$, but the extra point of $X^*$ is in the closure in $X^*$ of each of them. – Brian M. Scott Jan 29 '13 at 21:59
I have made that kind of mistake before. Maybe after making it this time I'll remember not to make it the next time! Hopefully. Thanks for the answer. – Alex Petzke Jan 30 '13 at 2:00

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.