Corollary

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

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

Let $f$ be a lower semi-continuous function on a non-empty quasi-compact space $X$ . If $f(x) > -\infty$ for all $x \in X$ , then $f$ is bounded below in $X$ .

Note that this theorem and the corresponding theorem for upper semi-continuous functions include Weierstrass's theorem as a particular case (no. 1, Theorem 1).

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

Cited by

Nothing.

Cites

no. 1, Theorem 1