Exercise 32.3

Question

Show that every locally compact Hausdorff space is regular.

Amit Awasthi · Accepted Answer

\begin{proof}
  Let $X$ be a locally compact Hausdorff space; in particular it is $T_1$.
  Let $x \in X$ with neighborhood $U 
i x$. By Theorem $29.2$, there exists a neighborhood $V 
i x$ such that $\overline{V} \subseteq U$. By Lemma $31.1(a)$, $X$ is then regular.
\end{proof}

Exercise 32.3

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Exercise 32.3

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