Bourbaki

Proposition 9

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

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

Let ff be a real-valued function defined on a product set X1×X2X_1 \times X_2. If A1,A2A_1, A_2 are any non-empty subsets of X1,X2X_1, X_2 respectively, then

sup⁡(x1,x2)∈A1×A2f(x1,x2)=sup⁡x1∈A1(sup⁡x2∈A2f(x1,x2))=sup⁡x2∈A2(sup⁡x1∈A1f(x1,x2)). \sup_{(x_1, x_2) \in A_1 \times A_2} f(x_1, x_2) = \sup_{x_1 \in A_1} (\sup_{x_2 \in A_2} f(x_1, x_2)) = \sup_{x_2 \in A_2} (\sup_{x_1 \in A_1} f(x_1, x_2)).

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

Cited by

Cites

Nothing.

Transcribed from the printed volume. Not translated.