In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has a ...

, a

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

of a Polish space has the perfect set property if it is either

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

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, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every

uncountable set In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger t ...

of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that

closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...

s of a Polish space ''X'' have the perfect set property in a particularly strong form: any closed subset of ''X'' can be written uniquely as the disjoint union of a perfect set and a countable set. 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 In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...

. As a consequence, if a subset

S \subset X

of a Polish space

X

is such that its derived sets eventually reach the empty set, that is,

S^ = \emptyset

for some ordinal

\alpha

, then

S

is countable. The axiom of choice implies the existence of sets of reals that do not have the perfect set property, such as Bernstein sets. However, in Solovay's model, which satisfies all axioms of ZF but not the axiom of choice, every set of reals has the perfect set property, so the use of the axiom of choice is necessary. Every analytic set has the perfect set property. It follows from the existence of sufficiently large cardinals that every projective set has the perfect set property.

Generalizations

Let

\omega_1

be the least uncountable ordinal. In an analog of Baire space derived from the

\omega_1

-fold

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

\omega_1

with itself, any closed set is the disjoint union of an

\omega_1

-perfect set and a set of

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

\leq\aleph_1

, where

\omega_1

-closedness of a set is defined via a topological game in which members of

\omega_1^

are played.J. Väänänen,
A Cantor-Bendixson theorem for the space

\omega_1^

. Fundamenta Mathematicae vol. 137, iss. 3, pp.187--199 (1991).

References

Citations

{{reflist Descriptive set theory