Remark
Tag 0BAT, permanent. Cite it as [Bourbaki, Tag 0BAT].
General Topology, chapter I, § 10. Proper mappings
If X is locally compact but not compact, then a closed equivalence relation R on X need not be Hausdorff (Chapter IX, § 4, Exercise 14); and even if R is Hausdorff, X/R need not be locally compact (Exercise 17). However, we have the following criterion:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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