Proposition 8

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

General Topology, chapter IV, § 5. Real-valued functions

Let $f$ be a real-valued function defined on $X$ , and let $(A_i)_{i \in I}$ be a non-empty family of non-empty subsets of $X$ ; then

\sup_{x \in \bigcup_{i \in I} A_i} f(x) = \sup_{i \in I} (\sup_{x \in A_i} f(x)).

Let $f$ be a real-valued function defined on a product set $X_1 \times X_2$ . If $A_2$ is a non-empty subset of $X_2$ we shall denote by $\sup_{x_1 \in A_2} f(x_1, x_2)$ the least upper bound in $A_2$ of the real-valued function $x_2 \to f(x_1, x_2)$ defined on $X_2$ . From Proposition 8 we deduce in particular:

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

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