Exercise 34.3

Question

Let $X$ be a compact Hausdorff space. Show that $X$ is metrizable if and only if $X$ has a countable basis.

Amit Awasthi · Accepted Answer

\begin{proof}
  $\Rightarrow$. $X$ is compact and metrizable, hence second countable
  by \href{exercise_30-4}{Exercise 30.4}.
  \par $\Leftarrow$. $X$ is compact and Hausdorff, and so $X$ is regular by Theorem $32.3$. $X$ is second countable as well, and so by the Urysohn metrization theorem (Theorem 34.1), $X$ is metrizable.
\end{proof}

Exercise 34.3

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Exercise 34.3

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