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of a perfect set and a countable set. In particular, every uncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set and a countable
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but not the axiom of choice, every set of reals has the perfect set property, so the use of the axiom of choice is necessary. Every
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subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a
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implies the existence of sets of reals that do not have the perfect set property, such as
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have the perfect set property in a particularly strong form: any closed subset of
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form a Polish space, a set of reals with the perfect set property cannot be a
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has the perfect set property. It follows from the existence of sufficiently
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464:". Fundamenta Mathematicae vol. 137, iss. 3, pp.187--199 (1991).
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As nonempty perfect sets in a Polish space always have the
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with itself, any closed set is the disjoint union of an
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456:{\displaystyle \omega _{1}^{\omega _{1}}}
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331:-closedness of a set is defined via a
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389:Classical Descriptive Set Theory
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67:cardinality of the continuum
324:{\displaystyle \omega _{1}}
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271:-perfect set and a set of
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385:Kechris, Alexander S.
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182:. In an analog of
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41:Polish space
29:mathematical
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273:cardinality
184:Baire space
139:that every
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60:perfect set
378:References
69:, and the
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410:Citations
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286:ℵ
282:≤
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51:or has a
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31:field of
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387:(1995),
304:, where
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27:In the
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39:of a
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151:Let
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