In 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). Note that having the perfect set property is not the same as being a perfect set.
As nonempty perfect sets in a Polish space always have the cardinality of the continuum, a set 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 set C may be written uniquely as the disjoint 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. Every analytic set has the perfect set property. It follows from sufficient large cardinals that every projective set has the perfect set property.
Famous quotes containing the words perfect, set and/or property:
“Good as is discourse, silence is better, and shames it. The length of the discourse indicates the distance of thought betwixt the speaker and the hearer. If they were at a perfect understanding in any part, no words would be necessary thereon. If at one in all parts, no words would be suffered.”
—Ralph Waldo Emerson (1803–1882)
“She leet no morsel from hir lippes falle,
Ne wette hir fyngres in hir sauce depe.
Wel koude she carie a morsel and wel kepe
That no drope ne fille upon hire brest.
In curteisie was set ful muchel hir lest.”
—Geoffrey Chaucer (1340?–1400)
“No man is by nature the property of another. The defendant is, therefore, by nature free.”
—Samuel Johnson (1709–1784)