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  "body": "\n## Definition\n\nLet $\\kappa$ be an uncountable [[regular cardinal|ZCE333FAWC]], and let $S \\subseteq \\kappa$.\nThe set $S$ is stationary in $\\kappa$ if $S \\cap C \\ne \\varnothing$ for each [[club set|GWS5QXH7WW]] $C$ in $\\kappa$.\n\n## Remarks\n\nExample: each club set $D$ in $\\kappa$ is stationary.\nCheck: for each club $C$, the intersection $C \\cap D$ is club, hence unbounded, hence nonempty.\nExample: the set $\\set{\\alpha \u003c \\omega_2 : \\operatorname{cf}(\\alpha) = \\omega}$ is stationary in $\\omega_2$.\nCheck: each club $C$ in $\\omega_2$ contains the supremum of its first $\\omega$ elements, and this supremum has cofinality $\\omega$.\nNon-example: a bounded set $S \\subseteq \\kappa$, with $S \\subseteq \\beta$ for some $\\beta \u003c \\kappa$, misses the club set $\\set{\\alpha : \\beta \\le \\alpha \u003c \\kappa}$.\nThe sets that are not stationary form a $\\kappa$-complete ideal, the nonstationary ideal; it is closed under diagonal unions.\nThe same definition is used for a limit ordinal of uncountable cofinality in place of $\\kappa$.\nBy a theorem of Solovay, each stationary subset of $\\kappa$ is the union of $\\kappa$ pairwise disjoint stationary sets.\n",
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