Section 34: Problem 3 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
A compact metric space is second-countable (exercise 4 of §30). A second-countable compact Hausdorff space is second-countable and normal (Theorem 32.3), therefore, metrizable (Theorem 34.1).