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- Jan 26, 2012

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**Problem**: A space $X$ is

**locally metrizable**if each point $x\in X$ has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space $X$ is metrizable if and only if it is locally metrizable.

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**Hint**:

Show that $X$ is a finite union of open subspaces, each of which has a countable basis.

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