Show that a compact Hausdorff space is is metrizable iff it is second-countable.
Solution
1. (\(\Rightarrow\)) If \(X\) is compact and metrizable, then \(X\) is second-countable.
a compact metric space is separable (totally bounded), and a separable metric space is second-countable.
2. (\(\Leftarrow\)) Suppose \(X\) is compact Hausdorff and second-countable.
assume the hypotheses.
3. A second-countable compact Hausdorff space is regular (compact Hausdorff implies normal, hence regular).
compact Hausdorff spaces are normal.
4. By the Urysohn metrization theorem, a second-countable regular space is metrizable.
Urysohn metrization theorem.
5. Hence \(X\) is metrizable.
3 and 4.
6. Q.E.D.
1 and 5.