Corollary

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

General Topology, chapter IV, § 6. Continuous and semi-continuous real-valued functions

A subset $A$ of a topological space $X$ is open (resp. closed) in $X$ if and only if its characteristic function () $\varphi_A$ is lower (resp. upper) semi-continuous on $X$ .*

For $\overline{f}_A([k, +\infty])$ is empty for $k \geq 1$ , is equal to $A$ for $0 \leq k < 1$ and is equal to $X$ for $k < 0$ .

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

Cited by

Proposition 5, § 1

Cites

Nothing.