Remark 3

Tag 0BD1, permanent. Cite it as [Bourbaki, Tag 0BD1].

General Topology, chapter II, § 1. Uniform spaces

If $X$ is not empty, then axiom (U_I) implies that no set of $\mathfrak{U}$ is empty, and therefore $\mathfrak{U}$ is a filter on $X \times X$ . There is only one uniformity on the empty set, namely $\mathfrak{U} = \{ \emptyset \}$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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