Definition 1
Tag 0BBG, permanent. Cite it as [Bourbaki, Tag 0BBG].
General Topology, chapter I, § 11. Connectedness
A topological space X is said to be connected if it is not the union of two disjoint non-empty open sets.
An equivalent definition is obtained by replacing the words "open sets" by "closed sets". X is connected if and only if the only subsets of X which are both open and closed are the empty set and the whole space X.
If X is connected and if A, B are two non-empty open (resp. closed) subsets such that A \cup B = X, then A \cap B \neq \emptyset.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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