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  "body": "\n## Definition\n\nLet $A$ be a nonempty set and let $X \\subseteq A^{\\N}$ be a set of infinite [[sequences|0G9EKM0E15]] of elements of $A$.\nThe game $G(A, X)$ is played by two players, I and II.\nPlayer I chooses $a_0 \\in A$, then player II chooses $a_1 \\in A$, then player I chooses $a_2 \\in A$, and so on forever.\nPlayer I wins the play $x = (a_0, a_1, a_2, \\dots)$ if $x \\in X$, and player II wins it otherwise.\n\n1. A *strategy* for player I is a map $\\sigma$ from the finite sequences of elements of $A$ of even length to $A$.\nA strategy for player II is a map $\\tau$ from the finite sequences of odd length to $A$.\n2. A play *follows* $\\sigma$ if $a_{2n} = \\sigma(a_0, \\dots, a_{2n-1})$ for all $n \\ge 0$, and similarly for $\\tau$ and the odd positions.\n3. A strategy for a player is *winning* if that player wins every play that follows it.\n\nThe game $G(A, X)$, and the set $X$, is *determined* if player I or player II has a winning strategy.\n\n## Remarks\n\nExample: let $A = \\N$ and $X = \\set{x \\in \\N^{\\N} : x_0 = 0}$.\nThe strategy of player I that plays $0$ at the first move, and anything later, is winning, so $X$ is determined.\n\nNon-example: under the axiom of choice there is a set $X \\subseteq \\set{0, 1}^{\\N}$ for which neither player has a winning strategy.\nIt is built by a diagonal argument against a well-ordering of all strategies.\n\nAt most one player can have a winning strategy: two winning strategies $\\sigma$ and $\\tau$ would produce a single play that both players win.\nThe case $A = \\N$ is the main one; then $A^{\\N}$ is the [[Baire space|P2WS16HKD7]] with its product topology.\n",
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