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  "body": "\n## Definition\n\nLet $X$ be a [[Polish space|584GXXY73M]], and let $\\mathcal{N} = \\N^\\N$ be the [[Baire space|P2WS16HKD7]].\nA subset $A \\subseteq X$ is *analytic* if $A = \\emptyset$ or there is a [[continuous map|0VGFHTM75Y]] $f \\colon \\mathcal{N} \\to X$ with $f(\\mathcal{N}) = A$.\nThe class of analytic subsets of $X$ is written $\\Sigma^1_1$.\n\n## Remarks\n\nEvery closed subset $F$ of $X$ is analytic: if $F$ is nonempty, it is a Polish space, and every nonempty Polish space is a continuous image of $\\mathcal{N}$.\nMore generally every Borel subset of $X$ is analytic.\nNon-example by counting: in $X = \\R$ there are only $2^{\\aleph_0}$ analytic sets.\nIndeed, each nonempty one is the image of a continuous map on $\\mathcal{N}$, and such a map is determined by its values on a countable dense set.\nSince $\\R$ has $2^{2^{\\aleph_0}}$ subsets, some subset of $\\R$ is not analytic.\nThe following are equivalent forms of the definition.\n$A$ is the projection to $X$ of a closed subset of $X \\times \\mathcal{N}$.\n$A$ is the projection to $X$ of a Borel subset of $X \\times Y$ for some Polish space $Y$.\n$A$ is empty or a continuous image of a Polish space.\nComplements of analytic sets are called coanalytic, and they form the class $\\Pi^1_1$.\n",
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