Polish space
Encyclopedia
In the mathematical discipline of general topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...

, a Polish space is a separable completely metrizable
Completely metrizable space
In mathematics, a completely metrizable space is a topological space for which there exists at least one metric d on X such that is a complete metric space and d induces the topology T...

 topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

; that is, a space homeomorphic to a complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

 metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

 that has a countable dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

 subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians — Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are primarily studied today because they are the primary setting for descriptive set theory
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...

, including the study of Borel equivalence relation
Borel equivalence relation
In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borel subset of X × X....

s.

Common examples of Polish spaces are the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

, any separable Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

, the Cantor space
Cantor space
In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the" Cantor space...

, and Baire space
Baire space (set theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called “reals.” It is often denoted B, N'N, or ωω...

. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval (0, 1) is Polish.

Between any two uncountable Polish spaces, there is a Borel isomorphism
Borel isomorphism
In mathematics, Borel isomorphism is a Borel measurable bijective function from one Polish space to another Polish space. Borel isomorphisms are closed under composition and under taking of inverses. The set of Borel isomorphisms from a Polish space to itself apparently forms a group under...

; that is, a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

 that preserves the Borel structure. In particular, every uncountable Polish space has 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 ....

.

Lusin space
Lusin space
In mathematics, a Lusin space or Luzin space, named for N. N. Luzin, may mean:*A Lusin space in measure theory and probability, the image of a Polish space under a bijective continuous map...

s
, Suslin spaces, and Radon space
Radon space
In mathematics, a Radon space, named after Johann Radon, is a separable metric space such that every Borel probability measure on M is inner regular. Since a probability measure is globally finite, and hence a locally finite measure, every probability measure on a Radon space is also a Radon...

s
are generalizations of Polish spaces.

Properties

  1. (Alexandrov
    Pavel Sergeevich Alexandrov
    Pavel Sergeyevich Alexandrov , sometimes romanized Aleksandroff or Aleksandrov was a Soviet Russian mathematician...

    's theorem) If X is Polish then so is any Gδ
    G-delta set
    In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...

     subset of X.
  2. (Cantor–Bendixson theorem) If X is Polish then any closed subset of X can be written 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 subset and a countable subset.


The converse of Alexandrov's theorem is true as well: if a subspace S of a Polish space X is Polish, then it is a Gδ subset of X.

Characterization

There are numerous characterizations that tell when a second countable topological space is metrizable, such as Urysohn's metrization theorem. The problem of determining whether a metrizable space is completely metrizable is more difficult. Topological spaces such as the open unit interval (0,1) can be given both complete metrics and incomplete metrics generating their topology.

There is a characterization of complete separable metric spaces in terms of a game known as the strong Choquet game
Topological game
A topological game is an infinite positional game of perfect information played between two players on a topological space. Players choose objects with topological properties such as points, open sets, closed sets and open coverings. Time is generally discrete, but the plays may have transfinite...

. A separable metric space is completely metrizable if and only if the second player has a winning strategy in this game.

A second characterization follows from Alexandrov's theorem. It states that a separable metric space is completely metrizable if and only if it is a subset of its completion in the original metric.

Polish metric spaces

Although Polish spaces are metrizable, they are not in and of themselves metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

s; each Polish space admits many complete metrics giving rise to the same topology, but no one of these is singled out or distinguished. A Polish space with a distinguished complete metric is called a Polish metric space. An alternative approach, equivalent to the one given here, is first to define "Polish metric space" to mean "complete separable metric space", and then to define a "Polish space" as the topological space obtained from a Polish metric space by forgetting
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...

 the metric.

Lusin spaces

A Lusin space is a topological space such that some weaker topology makes it into a Polish space.

There are many ways to form Lusin spaces. In particular:
  • Every Polish space is Lusin.
  • A subspace of a Lusin space is Lusin if and only if it is a Borel set.
  • Any countable union or intersection of Lusin subspaces of a Hausdorff space is Lusin.
  • The product of a countable number of Lusin spaces is Lusin.
  • The disjoint union of a countable number of Lusin spaces is Lusin.

Suslin spaces

A Suslin space is the image of a Polish space under a continuous mapping. So every Lusin space is Suslin.
In a Polish space, a subset is a Suslin space if and only if it is a Suslin set
Suslin set
The concept of a Suslin set was first used by Mikhail Yakovlevich Suslin when he was researching the properties of projections of Borel sets in \R^2 onto the real axis...

 (an image of the Suslin operation).

Radon spaces

A Radon space is a topological space such that every finite Borel measure is inner regular (so a Radon measure
Radon measure
In mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...

). Every Suslin space is Radon.

Polish groups

A Polish group is a topological group G regarded as a topological space which is itself a Polish space. A remarkable fact about Polish groups is that Baire-measurable (i.e., the preimage of any open set has the property of Baire) homomorphisms between them are automatically continuous. (Pettis in B. J. Pettis, ‘On continuity and openness of homomorphisms in topological groups’, Ann. of Math. vol. 51 (1950) 293–308, MR 38358)
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