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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $A \\subseteq X$.\nThe set $A$ has the *Baire property* if there is an [[open set|02BH3MCBQB]] $U \\subseteq X$ such that the [[symmetric difference|0ME21QN125]] $A \\mathbin{\\triangle} U$ is [[meagre|0XGCZ4GDZ6]] in $X$.\n\n## Remarks\n\nExample: every closed set $F \\subseteq X$ has the Baire property.\nTake $U$ to be the [[interior|0RZH6W98HK]] of $F$.\nThen $F \\mathbin{\\triangle} U = F \\setminus U$ is closed and has empty interior, so it is [[nowhere dense|0VQYCCQVJ5]] and hence meagre.\n\nNon-example: a Vitali set $V \\subseteq \\R$ contains exactly one point of each coset of $\\Q$ in $\\R$.\nSuch a set does not have the Baire property; its construction uses the axiom of choice.\n\nThe sets with the Baire property form a sigma-algebra that contains the open sets, so every Borel set has the Baire property.\nEquivalently, $A$ has the Baire property if $A = G \\cup M$ for a $G_\\delta$ set $G$ and a meagre set $M$.\n",
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