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When does $$\text{sup}_{x\in A} \text{sup}_{y\in B}f(x,y) = \text{sup}_{y\in B}\text{sup}_{x\in A} f(x,y)$$ fail?

Further can we also have then that $$\text{sup}_{x \in A, y\in B} f(x,y)$$ is greater than both of the expressions above?

(By principle of iterated supremum, we need at least that $\text{sup}_{x \in A, y\in B} f(x,y)=\infty$).

Tony
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  • Never for nonempty $A,B$, as long as $f$ takes its values in a set where nonempty subsets have a supremum (such as $[0,\infty]$). – Aphelli Apr 09 '20 at 13:09
  • @Mindlack Okay, but that sounds close to the principle of iterated supremum. I want to see when it fails... – Tony Apr 09 '20 at 14:32

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