Section 30: Problem 6 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
is separable but not second-countable (proved in the text). The ordered square is compact (it is a linear continuum and a closed interval) but not separable (take the uncountable collection of vertical open intervals, each must have at least one dense point). Note that both are examples of first-countable but not second-countable spaces.