Given any $f: X \times Y \rightarrow \mathbb{R} \cup \{ \infty, -\infty\}$, I was wondering
- if it is true that $\inf_{x \in X} \sup_{y \in Y} f(x,y) \geq \sup_{y \in Y} \inf_{x \in X} f(x,y)$?
- when it is true that $\inf_{x \in X} \sup_{y \in Y} f(x,y) = \sup_{y \in Y} \inf_{x \in X} f(x,y)$?
- if the answers to the above will be different if $\inf$ and/or $\sup$ be replaced with $\min$ and/or $\max$?
- if the answers to the above will be different if the codomain of $f$ is any totally ordered set? Suppose all exist.
Thank!