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  "body": "\n## Definition\n\nA [[topological space|0CNQXVBHCX]] $X$ is a Baire space when it has the property that follows.\nLet $(F_n)_{n \\in \\mathbb{N}}$ be a [[countable|0ZG61TFBES]] family of [[closed subsets|0G5AV9Q4RR]] of $X$ with $\\operatorname{int} F_n = \\emptyset$ for each $n$.\nThen also $\\operatorname{int} \\bigcup_n F_n = \\emptyset$.\n\n## Remarks\n\nHere $\\operatorname{int}$ is the [[interior|0C38S0KMJK]].\nEquivalently, in a Baire space each [[countable intersection of dense open sets is dense|0X2T63GFQQ]].\nIn a Baire space, each [[meagre set|0FQPYAY9PJ]] has empty interior. Thus a nonempty Baire space is not meagre in itself.\nEach complete metric space is a Baire space by the [[Baire category theorem|0A7KGAYHGV]]. The space $\\mathbb{Q}$ is not a Baire space.\n\n## Explanations\n\n### L1 human\n\nIn a Baire space, countably many thin closed sets can never fill up an open set.\nThis says that the space is large. It lets you prove that some objects exist: you show that the bad cases form a small set.\n",
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