Definition 4
Tag 0C2Z, permanent. Cite it as [Bourbaki, Tag 0C2Z].
General Topology, chapter IX, § 2. Metric spaces and metrizable spaces
A metrizable space is said to be of countable type (or separable) if its topology has a countable base.
Clearly every subspace of a metrizable space of countable type is again of countable type. The definition of the base of the topology of a product space (Chapter I, § 4, no. 1) and Corollary 2 to Theorem 1 of no. 4 show that the product of a countable family of metrizable spaces of countable type is a metrizable space of countable type. Again, the sum of a countable family of metrizable spaces of countable type is a metrizable space of countable type (no. 5).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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