Section 30: Problem 4 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
As in the hint: take a covering by
and its finite subcovering. The union over all
is a countable basis. Another way to show this is as follows: compact metric implies Lindelöf metric implies second-countable.