Proposition 7
Tag 0BGK, permanent. Cite it as [Bourbaki, Tag 0BGK].
General Topology, chapter II, § 4. Relations between uniform spaces and compact spaces
Let X be a compact space and let R be the equivalence relation on X whose classes are the components of X. Then the quotient space X/R is compact and totally disconnected.
We know from Chapter I, § II, no. 5, Proposition 9 that X/R is totally disconnected; hence we have to show that X/R is Hausdorff (Chapter I, § 10, no. 4, Proposition 8.) Let A and B be two distinct components of X. By Proposition 6 there is a symmetric entourage U of X such that no point of A can be joined to any point of B by a U-chain. The set V (resp. W) of points of X which can be joined to a point x ∈ A (resp. y ∈ B) by a U-chain is both open and closed in X (Proposition 5) and contains A (resp. B); these sets are therefore open neighbourhoods of A and B respectively, are saturated with respect to R and do not intersect. This completes the proof.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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