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perfect set property wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 generalizations 2 references toggle references subsection 2 1 citations toggle the table of contents perfect set property 2 languages français 日本語 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia property in descriptive set theory in the mathematical field of descriptive set theory a subset of a polish space has the perfect set property if it is either countable or has a nonempty perfect subset 1 note that having the perfect set property is not the same as being a perfect set as nonempty perfect sets in a polish space always have the cardinality of the continuum and the reals form a polish space a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis stated in the form that every uncountable set of reals has the cardinality of the continuum the cantor bendixson theorem states that closed sets of a polish space x displaystyle x have the perfect set property in a particularly strong form any closed subset of x displaystyle x can be written uniquely as the disjoint union 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 open set as a consequence if a subset s x displaystyle s subseteq x of a polish space x displaystyle x is such that its derived sets eventually reach the empty set that is s α displaystyle s alpha varnothing for some ordinal α displaystyle alpha then s displaystyle s is countable the axiom of choice implies the existence of sets of reals that do not have the perfect set property such as bernstein sets however in solovay s model which satisfies all axioms of zf 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 analytic set has the perfect set property it follows from the existence of sufficiently large cardinals that every projective set has the perfect set property generalizations edit let ω 1 displaystyle omega _ 1 be the least uncountable ordinal in an analog of baire space derived from the ω 1 displaystyle omega _ 1 fold cartesian product of ω 1 displaystyle omega _ 1 with itself any closed set is the disjoint union of an ω 1 displaystyle omega _ 1 perfect set and a set of cardinality ℵ 1 displaystyle leq aleph _ 1 where ω 1 displaystyle omega _ 1 closedness of a set is defined via a topological game in which members of ω 1 ω 1 displaystyle omega _ 1 omega _ 1 are played 2 references edit kechris alexander s 1995 classical descriptive set theory graduate texts in mathematics vol 156 new york springer isbn 978 0 387 94374 9 väänänen jouko 1991 a cantor bendixson theorem for the space ω 1 ω 1 displaystyle omega _ 1 omega _ 1 fundamenta mathematicae 137 3 187 199 zbl 0732 03041 citations edit kechris 1995 p 150 väänänen 1991 retrieved from https en wikipedia org w index php title perfect_set_property oldid 1371573131 category descriptive set theory hidden categories articles with short description short description matches wikidata this page was last edited on 27 august 2026 at 05 00 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents perfect set property 2 languages add topic