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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *Polish* if:\n\n1. $X$ is [[separable|0Z2P41SVDP]]: it has a countable dense subset.\n2. $X$ is completely metrizable: there is a [[metric|0VKD1TP7F3]] $d$ on $X$ that induces the topology of $X$ and makes $(X, d)$ a [[complete metric space|0BY52JW6NK]].\n\n## Remarks\n\nThe real line $\\R$ is Polish: the rationals are dense, and the usual metric is complete.\nThe open interval $(0, 1)$ is Polish, because the metric $d(x, y) = \\abs{\\tan(\\pi x - \\pi/2) - \\tan(\\pi y - \\pi/2)}$ induces its topology and is complete.\nThe rationals $\\Q$ are a non-example: they are separable and metrizable, but no complete metric induces their topology.\nIndeed, by the Baire category theorem a nonempty complete metric space is not a countable union of nowhere dense sets, while $\\Q$ is the countable union of its points.\nAn uncountable discrete space is another non-example: it is completely metrizable but not separable.\nA subset of a Polish space is Polish in the subspace topology if and only if it is a $G_\\delta$ set.\n",
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