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. In mathematics, descriptive set theory is the study of certain classes of well-behaved sets of real numbers, e. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics the term countable set is used to describe the size of a set, e. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In mathematics a derived set is a construction in point-set topology that consists of taking the set of limit points of a given subset S of a topological space X. The derived set of S is usually denoted by S′. A subset S of a topological space X is...

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. In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, an uncountable set is a set which is not countable. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

The Cantor-Bendixson theorem states that closed sets have the perfect set property in a particularly strong form; any closed set may be written uniquely as the disjoint union of a perfect set and a countable set. In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

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. In mathematics, the axiom of choice is an axiom of set theory. ... In mathematical logic and descriptive set theory, the analytical hierarchy is a second-order analogue of the arithmetical hierarchy. ... In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ... In mathematical logic and descriptive set theory, the analytical hierarchy is a second-order analogue of the arithmetical hierarchy. ...