Remark 2
Tag 0BG2, permanent. Cite it as [Bourbaki, Tag 0BG2].
General Topology, chapter II, § 4. Relations between uniform spaces and compact spaces
It can happen that there are several distinct uniformities compatible with the topology of a locally compact space.
For example, we have seen that on an infinite discrete space there is more than one distinct uniformity compatible with the discrete topology (§ 2, no. 2, Remark).
Nevertheless it is not the case that the uniqueness of the uniformity compatible with the topology of a uniformizable space is a property which characterizes compact spaces; there exist locally compact spaces which are not compact but whose uniformity is unique (Exercise 4).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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