In this paper, page 6, Itai Arieli and Yehuda Levy mention briefly using using Minimax Theorem of prove the determinacy of closed game in a more general setting.
The minimax theorem they mentioned can be founded here:
I can't understand why this is true. It seems to me with respect to payoff function $f: X \times Y \to [0,1]$, $X$ and $Y$ are two players' pure strategy space, which are subsets of $2^{A^{\omega}}$. What's the topology defined on pure strategy space to talk about Hausdorff condition, compactness, and semicontinuity?
Added: I found some clue in a cited paper by Eran Shmaya(see page 7-8 here).The pure stategy space is indeed compact, since it's a product of finite function spaces. But the definition of a game has a value is kind of weird to me, since it seems to me it can't be matched to Ky Fan's theorem.
In particular, I can't see why there's need to introduce mixed strategy space.
The compactness of mixed strategy space, as the space of all measures of pure strategy space, is implied by compactness of pure strategy space, which is a product of finite function spaces. Mixed strategy guarentees that payoff function is affine in variables, which matches conditions in the Ky Fan's theorem that $f$ is convex and concave in two varibles respectively. To show the l.s.c. of the original payoff function, it suffices to show the restriction of payoff function is l.s.c. with respect to player I's pure strategy space, given any player II's pure strategy, which is surely l.s.c., given closed winning set.