Perfect set property
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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.
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 may be written uniquely as the disjoint union of a perfect set P and a countable open set O. The set P is then called the perfect kernel of X.
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.
