Proposition 2
Tag 0B80, permanent. Cite it as [Bourbaki, Tag 0B80].
General Topology, chapter I, § 9. Compact spaces and locally compact spaces
Let X be a compact space and let A, B be two disjoint closed subsets of X. Then there exist two open sets U, V, such that U ∩ V = ∅ and A ⊂ U and B ⊂ V.
Suppose the conclusion is false. If every neighbourhood U of A meets every neighbourhood V of B, then the sets U ∩ V form a filter base B on X, which therefore has a cluster point x ∈ X. Now x must lie in A, since if y is any point of X not in A there is a neighbourhood of y and a neighbourhood of A which do not intersect, since X is regular, and thus y cannot be a cluster point of B. Similarly x must lie in B and we have a contradiction.
This proposition has important consequences which will be examined in Chapter IX, § 4.
The non-Hausdorff quasi-compact space X of Example 2 of no. 1 does not satisfy axiom (O_{III}), nor a fortiori the property stated in Proposition 2, since any two non-empty open sets of this space always intersect.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.