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  "body": "\n## Definition\n\nA *Polish space* is a [[topological space|0CQHFPGJ45]] $(X, \\tau)$ that is [[separable|0Z2P41SVDP]] and has the following property.\nThere is a metric $d$ on $X$ that induces $\\tau$ such that $(X, d)$ is a [[complete metric space|0BY52JW6NK]].\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]].\nIt is a *standard Borel space* if $X$ has a topology $\\tau$ with these two properties.\nThe space $(X, \\tau)$ is a Polish space, and $\\Sigma$ is the [[Borel σ-algebra|0WQXY32Q8D]] of $\\tau$.\n\n## Remarks\n\nThe real line $\\R$ with its Borel σ-algebra is a standard Borel space, since the usual metric on $\\R$ is complete and $\\Q$ is a countable dense subset.\nA countable set with its power set is a standard Borel space, using the discrete metric.\nLet $\\Sigma$ be the family of subsets $A \\subseteq \\R$ such that $A$ or $\\R \\setminus A$ is countable.\nThen $(\\R, \\Sigma)$ is a measurable space but not a standard Borel space.\nIndeed, the Borel σ-algebra of a Polish space is generated by countably many sets.\nThese are the open balls with rational radius around the points of a countable dense set.\nThe σ-algebra $\\Sigma$ is not generated by countably many sets.\nEquivalently, a standard Borel space is a measurable space isomorphic to a Polish space with its Borel σ-algebra.\nKuratowski's theorem says that a standard Borel space is countable or has the cardinality of $\\R$, and that two standard Borel spaces of the same cardinality are isomorphic.\n",
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