Perfect set property

Summary

A subset of a Polish space has the perfect set property if it is either countable or contains a nonempty perfect set. A nonempty perfect subset of a Polish space has the cardinality of the continuum. So an uncountable set with this property is as large as the whole real line. Thus the continuum hypothesis holds for every class of sets that has the property. Closed sets, Borel sets and analytic sets have the perfect set property, but the axiom of choice gives sets of reals without it.