Example 6

Tag 0B38, permanent. Cite it as [Bourbaki, Tag 0B38].

General Topology, chapter I, § 5. Open mappings and closed mappings

Let $X, X'$ be two topological spaces and $f$ a continuous, but not bicontinuous, bijection of $X$ onto $X'$ . Then the inverse bijection $g : X' \to X$ is an open and closed mapping of $X'$ onto $X$ , but is not continuous.

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

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