Theorem 2 in Ky Fan(1952) is a powerful tool in zero-sum games, which states:
Let $X$ be a compact Hausdorff space and $Y$ an arbitary set (not topologized). Let $f$ be a real-valued function on $X \times Y$ such that, for every $y \in Y$, $f(x,y)$ is lower semicontinuous on $X$, if $f$ is convex on $X$ and concave on $Y$, then $$\min_{x \in X} \sup_{y \in Y} f(x,y) = \sup_{y \in Y}\min_{x \in X}f(x,y)$$
The original proof of Ky Fan eludes me. I suspect it's because I don't grasp some standard material which may be covered in a textbook. Because this is an old paper, I wonder whether there exists some more polished proof of this theorem. Is there such a textbook, or some readable notes that explain this theorem as some others do to Brower fixed point theorem? I'm also interested in the generalization of this theorem in the sense that some conditions, say, compactness, lower semicontinuity, can be relaxed, though not removed.
Added: I know some basics of general topology which I think suffices to aid understanding of this theorem. But I only know some definitions of convex optimization. Should I do some exercises on some relevant chapter of convex optimization?