More General Topology

This probably doesn’t deserve a § all to itself, but it will be handy to know it in what follows. Recall that a set S in a metric (or topological) space X is dense-in-itself 1 if every neighborhood of each s0 ∈ S contains points s ∈ S different from s0. One says that S is perfect if it is closed and dense-in-itself. A topological (or metric) space is separable if it has a countable subset whose closure is the whole space. Recall that for metric spaces, being separable is equivalent to being second countable: there is a countable family B of open sets such that every open set is the union of the elements of B that it contains. This property is inherited by subsets of these spaces (in their relativized metric).