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- Perfect_set_property abstract "In 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 (Kechris 1995, p. 150). 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, a set 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 have the perfect set property in a particularly strong form; any closed set C may be written uniquely as the disjoint union of a perfect set P and a countable set S. Thus it follows that every closed subset of a Polish space has the perfect set property. 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.It follows from the axiom of choice that there are sets of reals that do not have the perfect set property. Every analytic set has the perfect set property. It follows from sufficient large cardinals that every projective set has the perfect set property.".
- Perfect_set_property wikiPageID "2221109".
- Perfect_set_property wikiPageLength "1653".
- Perfect_set_property wikiPageOutDegree "20".
- Perfect_set_property wikiPageRevisionID "488354535".
- Perfect_set_property wikiPageWikiLink Analytic_set.
- Perfect_set_property wikiPageWikiLink Axiom_of_choice.
- Perfect_set_property wikiPageWikiLink Cardinality_of_the_continuum.
- Perfect_set_property wikiPageWikiLink Category:Descriptive_set_theory.
- Perfect_set_property wikiPageWikiLink Closed_set.
- Perfect_set_property wikiPageWikiLink Continuum_hypothesis.
- Perfect_set_property wikiPageWikiLink Countable_set.
- Perfect_set_property wikiPageWikiLink Counterexample.
- Perfect_set_property wikiPageWikiLink Descriptive_set_theory.
- Perfect_set_property wikiPageWikiLink Disjoint_union.
- Perfect_set_property wikiPageWikiLink Empty_set.
- Perfect_set_property wikiPageWikiLink Large_cardinal.
- Perfect_set_property wikiPageWikiLink Nonempty.
- Perfect_set_property wikiPageWikiLink Perfect_set.
- Perfect_set_property wikiPageWikiLink Polish_space.
- Perfect_set_property wikiPageWikiLink Projective_hierarchy.
- Perfect_set_property wikiPageWikiLink Projective_set.
- Perfect_set_property wikiPageWikiLink Real_number.
- Perfect_set_property wikiPageWikiLink Springer-Verlag.
- Perfect_set_property wikiPageWikiLink Springer_Science+Business_Media.
- Perfect_set_property wikiPageWikiLink Subset.
- Perfect_set_property wikiPageWikiLink Uncountable_set.
- Perfect_set_property wikiPageWikiLinkText "Perfect set property".
- Perfect_set_property wikiPageWikiLinkText "perfect set property".
- Perfect_set_property wikiPageWikiLinkText "perfect set theorem".
- Perfect_set_property wikiPageWikiLinkText "perfect subset property".
- Perfect_set_property wikiPageWikiLinkText "perfect".
- Perfect_set_property hasPhotoCollection Perfect_set_property.
- Perfect_set_property wikiPageUsesTemplate Template:Citation.
- Perfect_set_property subject Category:Descriptive_set_theory.
- Perfect_set_property comment "In 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 (Kechris 1995, p. 150).".
- Perfect_set_property label "Perfect set property".
- Perfect_set_property sameAs m.06w_wd.
- Perfect_set_property sameAs Q7168098.
- Perfect_set_property sameAs Q7168098.
- Perfect_set_property wasDerivedFrom Perfect_set_property?oldid=488354535.
- Perfect_set_property isPrimaryTopicOf Perfect_set_property.