In Sect. 5.2, we have seen that the notions of countable compactness and sequential compactness coincide for topological spaces which have countable local bases. A more important but restricted class of spaces consists of the ones which have countable bases. The first section concerns with such spaces. There are two other properties, namely, separability and Lindelöfness which are described by using the notion of countability, and each of them guarantees the existence of countable bases in metrizable spaces. These are treated in the second section.