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 (Kechris 1995, p. 150).
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 can be written uniquely as the disjoint union of a perfect set and a countable set.
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.
It follows from the existence of sufficiently large cardinals that every projective set has the perfect set property.
In an analog of Baire space derived from the
with itself, any closed set is the disjoint union of an
-closedness of a set is defined via a topological game in which members of