Proposition 1
Tag 0B7Y, permanent. Cite it as [Bourbaki, Tag 0B7Y].
General Topology, chapter I, § 9. Compact spaces and locally compact spaces
Let X be a compact space, x a point of X. In order that a filter base B formed of closed neighbourhoods of x should be a fundamental system of neighbourhoods of x it is necessary and sufficient that the intersection of the sets of B consists of x alone.
The condition is necessary since X is Hausdorff (§ 8, no. 1, Proposition 1). It is sufficient, for it signifies that x is the only cluster point of B; hence B converges to x by the Corollary to Theorem 1 of no. 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.