Corollary
Tag 0BSR, permanent. Cite it as [Bourbaki, Tag 0BSR].
General Topology, chapter IV, § 6. Continuous and semi-continuous real-valued functions
The upper envelope of a family of continuous real-valued functions on a space is lower semi-continuous on .
In Chapter IX, § 1, no. 6, Proposition 5, we shall show that the converse of this proposition is true if is uniformizable (and only in this case): every lower semi-continuous function on a uniformizable space is the upper envelope of a family of continuous functions.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.