Section 30: Problem 17 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
It is not first countable for the reason exactly the same as in Exercise 1(b). Therefore, it is not second-countable. It is countable, therefore, separable and Lindelöf.