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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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453:". 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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74:of reals has the cardinality of the continuum.
445:{\displaystyle \omega _{1}^{\omega _{1}}}
413:A Cantor-Bendixson theorem for the space
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320:-closedness of a set is defined via a
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378:Classical Descriptive Set Theory
286:{\displaystyle \leq \aleph _{1}}
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132:has the perfect set property.
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56:cardinality of the continuum
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313:{\displaystyle \omega _{1}}
253:{\displaystyle \omega _{1}}
226:{\displaystyle \omega _{1}}
195:{\displaystyle \omega _{1}}
160:{\displaystyle \omega _{1}}
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260:-perfect set and a set of
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79:Cantor–Bendixson theorem
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374:Kechris, Alexander S.
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324:in which members of
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171:. In an analog of
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30:Polish space
18:mathematical
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262:cardinality
173:Baire space
128:that every
83:closed sets
49:perfect set
367:References
58:, and the
432:ω
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399:Citations
343:ω
333:ω
302:ω
275:ℵ
271:≤
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40:or has a
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464:Category
376:(1995),
293:, where
99:open set
42:nonempty
32:has the
169:ordinal
66:to the
45:perfect
16:In the
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202:-fold
26:subset
60:reals
28:of a
386:ISBN
140:Let
104:The
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