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Perfect set property

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. Note that having the perfect set property is not the same as being a perfect set.

Generalizations
Let \omega_1 be the least uncountable ordinal. In an analog of Baire space derived from the \omega_1-fold Cartesian product of \omega_1 with itself, any closed set is the disjoint union of an \omega_1-perfect set and a set of cardinality \leq\aleph_1, where \omega_1-closedness of a set is defined via a topological game in which members of \omega_1^{\omega_1} are played. == References ==
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