Definition 1
Tag 0CF0, permanent. Cite it as [Bourbaki, Tag 0CF0].
General Topology, chapter X, § 4. Approximation of continuous real-valued functions
If X is any set, a set H of mappings of X into a set Y is said to separate the elements of a subset A of X (or to be a separating set for the elements of A) if, given any two distinct elements x, y of A, there is a function f ∈ H such that f(x) ≠ f(y).
For example, if X is a completely regular space (Chapter IX, § 1, no. 5) then the set of all continuous mappings of X into [0, 1] separates the points of X.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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