Section 30: Problem 9 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
This is pretty much the same as for compact spaces: take any open covering, extend open sets to open sets in
, add
, find a countable subcovering. Now, the second part asks to show by an example that a closed subspace of a separable space need not be separable. Indeed,
is separable but its "inverse diagonal" is uncountable and discrete in the subspace topology.