According to Von Neumann's minimax theorem, I have
$$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$
for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) function $f(x,y)$.
Why is the compactness needed for this to hold?
Whatever function I choose from $X$ and $Y$, $f(x,y)$ will be concave in $x$ and convex in $y$, which suggests a saddle point, that implies minimax theorem.