Proposition 3

Tag 0BBO, permanent. Cite it as [Bourbaki, Tag 0BBO].

General Topology, chapter I, § 11. Connectedness

Let $A$ be a subset of a topological space $X$ . If $B$ is a connected subset of $X$ which meets both $A$ and $\complement A$ , then $B$ meets the frontier of $A$ .

For otherwise the intersections of $B$ with the interior and exterior of $A$ would be two open subsets of $B$ which form a partition of $B$ , and $B$ would not be connected.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Lemma 1, § 11
Exercise 13, § 5

Cites

Nothing.