Definition 1.1. Let (X, τ) be a topological space. A subset D ⊆ X is called dense if D ∩O 6= ∅ for every nonempty open set O ⊆ X. X is called separable if X has a countable dense subset. X is called metrizable if there is a metric d on X such that the topology τ is the same as the topology induced by the metric. The metric is called complete if every Cauchy sequence converges in X. Finally, X i...
