Perfect set property
Encyclopedia
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 real
s has the cardinality of the continuum.
The Cantor–Bendixson theorem states that closed set
s 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.
Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
, a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of a Polish space
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
has the perfect set property if it is either countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
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
Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....
, a set with the perfect set property cannot be a counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
to the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
, stated in the form that every uncountable set
Uncountable set
In mathematics, an uncountable set 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 than that of the set of all natural numbers.-Characterizations:There...
of real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s has the cardinality of the continuum.
The Cantor–Bendixson theorem states that closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
s 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
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a 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
Analytic set
In descriptive set theory, a subset of a Polish space X is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student .- Definition :There are several equivalent definitions of analytic set...
has the perfect set property. It follows from sufficient large cardinals that every projective set has the perfect set property.