Section 30: Problem 8 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 a metric space, therefore, it is first-countable and all other three properties are equivalent. In fact, they do not hold. We show that it is not second-countable. According to exercise 3, if it was second-countable, then every uncountable set would have a limit point. But if we take a set of all sequences of 0’s and 1’s, then it is uncountable but the distance between any two elements is 1.