- Perfect set property
In
descriptive set theory , asubset of aPolish space has the perfect set property if it is either countable or has anonempty 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 acounterexample to thecontinuum hypothesis , stated in the form that everyuncountable set of reals has the cardinality of the continuum.The Cantor-Bendixson theorem states that
closed set s of a Polish space "X" have the perfect set property in a particularly strong form; any closed set "C" may be written uniquely as thedisjoint 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. Everyanalytic set has the perfect set property. It follows from sufficientlarge cardinal s that everyprojective set has the perfect set property.References
* [http://www.reed.edu/~sollaa/samples/alexander.solla.thesis.pdf "The Borel Isomorphism Theorem"]
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