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:
“It would be difficult for me not to conclude that the most perfect type of masculine beauty is Satan,—as portrayed by Milton.”
—Charles Baudelaire (1821–1867)
“I set forth a humble and inglorious life; that does not matter. You can tie up all moral philosophy with a common and private life just as well as with a life of richer stuff. Each man bears the entire form of man’s estate.”
—Michel de Montaigne (1533–1592)
“No man acquires property without acquiring with it a little arithmetic, also.”
—Ralph Waldo Emerson (1803–1882)