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  "body": "\n## Summary\n\nA subset of a [[Polish space|584GXXY73M]] has the perfect set property if it is either countable or contains a nonempty [[perfect set|628242KN0T]].\nA nonempty perfect subset of a Polish space has the [[cardinality of the continuum|0WC28W9DNM]].\nSo an uncountable set with this property is as large as the whole real line.\nThus the [[continuum hypothesis|0MN2MER9WN]] holds for every class of sets that has the property.\nClosed sets, Borel sets and [[analytic sets|3RWWZYPAFV]] have the perfect set property, but the axiom of choice gives sets of reals without it.\n\n## Explanations\n\n### L1 human\n\nThe property says that a set of reals cannot be uncountable in a thin or scattered way.\nEither the set can be listed as a sequence, or it contains a whole copy of the Cantor set.\nFor example, the interval $[0, 1]$ is uncountable and is itself perfect: it is closed and none of its points is isolated.\nThe set of rational numbers is countable, so it has the property for the other reason.\nDescriptive set theory asks which definable sets have the property, and the answer depends on the axioms one adopts.\n",
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