Perfect set property (original) (raw)
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, 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.
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| dbo:abstract | En théorie descriptive des ensembles, un sous-ensemble d'un espace polonais a la propriété d'ensemble parfait s'il est soit dénombrable soit possède un sous-ensemble parfait non vide. Notons qu'avoir la propriété d'ensemble parfait n'est pas équivalent à être un ensemble parfait. Puisque tout espace polonais parfait non vide a toujours la puissance du continu, et que l'ensemble des réels forme un espace polonais, un ensemble de réels avec la propriété d'ensemble parfait ne peut être un contre-exemple à l'hypothèse du continu, statuant que tout ensemble de réels non dénombrable possède la puissance du continu. Le théorème de Cantor-Bendixson établit que les ensembles fermés d'un espace polonais X ont la propriété d'ensemble parfait sous une forme particulièrement forte : tout sous-ensemble fermé de X peut être écrit de manière unique comme union disjointe d'un ensemble parfait et d'un ensemble dénombrable. En particulier, tout espace polonais indénombrable possède la propriété d'ensemble parfait, et peut s'écrire comme l'union disjointe d'un ensemble parfait et d'un ensemble dénombrable. L'axiome du choix implique l'existence d'ensembles de réels qui n'ont pas la propriété d'ensemble parfait, tels que les (en). Cependant, dans le (en), qui satisfait tous les axiomes de ZF mais pas l'axiome du choix, tout ensemble de réels a la propriété d'ensemble parfait ; l'utilisation de l'axiome du choix est donc nécessaire. Tout (en) a la propriété d'ensemble parfait. Il suit de l'existence de cardinaux suffisamment grands que tout (en) a la propriété d'ensemble parfait. (fr) 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, 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 have the perfect set property in a particularly strong form: any closed subset of X may 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. 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. (en) |
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| rdfs: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). 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. (en) En théorie descriptive des ensembles, un sous-ensemble d'un espace polonais a la propriété d'ensemble parfait s'il est soit dénombrable soit possède un sous-ensemble parfait non vide. Notons qu'avoir la propriété d'ensemble parfait n'est pas équivalent à être un ensemble parfait. (fr) |
| rdfs:label | Propriété d'ensemble parfait (fr) Perfect set property (en) |
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