Section 32: 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
The only difference from the proof of the Theorem 32.1 is that we construct a countable covering using the Lindelöf property of the space instead of its countable basis.