Assertion: set's property | /r/HasProperty | perfect set
Facets:
Source sentences:
text: It also proves that every analytic set has the perfect set property.
text: But the perfect set property for co-analytic sets implies that the first uncountable cardinal, \(\aleph_1\), is a large cardinal in the constructible universe \(L\) (see Section 7), namely a so-called inaccessible cardinal (see Section 10), which implies that one cannot prove in ZFC that every co-analytic set has the perfect set property.
text: Other classical regularity properties are the Baire property (a set of reals has the Baire property if it differs from an open set by a meager set, namely, a set that is a countable union of sets that are not dense in any interval), and the perfect set property (a set of reals has the perfect set property if it is either countable or contains a perfect set, namely, a nonempty closed set with no isolated points).