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 topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.

- Properties
- Characterization
- Polish metric spaces
- Generalizations of Polish spaces
- Lusin spaces
- Suslin spaces
- Radon spaces
- Polish groups
- See also
- Notes
- References
- Further reading

Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and the Baire space. 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; that is, a bijection that preserves the Borel structure. In particular, every uncountable Polish space has the cardinality of the continuum.

** Lusin spaces **, ** Suslin spaces **, and ** Radon spaces ** are generalizations of Polish spaces.

- Every Polish space is second countable (by virtue of being separable metrizable).
- (Alexandrov's theorem) If
*X*is Polish then so is any*G*_{δ}subset of*X*.^{ [1] } - A subspace
*Q*of a Polish space*P*is Polish if and only if*Q*is the intersection of a sequence of open subsets of*P*. (This is the converse to Alexandrov's theorem.)^{ [2] } - (Cantor–Bendixson theorem) If
*X*is Polish then any closed subset of*X*can be written as the disjoint union of a perfect set and a countable set. Further, if the Polish space*X*is uncountable, it can be written as the disjoint union of a perfect set and a countable open set. - Every Polish space is homeomorphic to a G
_{δ}-subset of the Hilbert cube (that is, of*I*^{ℕ}, where*I*is the unit interval and ℕ is the set of natural numbers).^{ [3] }

The following spaces are Polish:

- closed subsets of a Polish space,
- open subsets of a Polish space,
- products and disjoint unions of countable families of Polish spaces,
- locally compact spaces that are metrizable and countable at infinity,
- countable intersections of Polish subspaces of a Hausdorff topological space,
- the set of irrational numbers with the topology induced by the standard topology of the real line.

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. 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.

Although Polish spaces are metrizable, they are not in and of themselves metric spaces; 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 the metric.

A topological space is a **Lusin space** if it is homeomorphic to a Borel subset of a compact metric space.^{ [4] }^{ [5] } Some stronger topology makes a Lusin into a Polish space.

There are many ways to form Lusin spaces. In particular:

- Every Polish space is Lusin
^{ [6] } - A subspace of a Lusin space is Lusin if and only if it is a Borel set.
^{ [7] } - Any countable union or intersection of Lusin subspaces of a Hausdorff space is Lusin.
^{ [8] } - The product of a countable number of Lusin spaces is Lusin.
^{ [9] } - The disjoint union of a countable number of Lusin spaces is Lusin.
^{ [10] }

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 (an image of the Suslin operation).^{ [11] }

The following are Suslin spaces:

- closed or open subsets of a Suslin space,
- countable products and disjoint unions of Suslin spaces,
- countable intersections or countable unions of Suslin subspaces of a Hausdorff topological space,
- continuous images of Suslin spaces,
- Borel subsets of a Suslin space.

They have the following properties:

- Every Suslin space is separable.

A **Radon space**, named after Johann Radon, is a topological 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 measure. In particular a separable complete metric space (*M*, *d*) is a Radon space.

Every Suslin space is Radon.

A **Polish group** is a topological group *G* that is also a Polish space, in other words homeomorphic to a separable complete metric space. There are several classic results of Banach, Freudenthal and Kuratowski on homomorphisms between Polish groups.^{ [12] } Firstly, the argument of Banach (1932 , p. 23) applies *mutatis mutandi* to non-Abelian Polish groups: if *G* and *H* are separable metric spaces with *G* Polish, then any Borel homomorphism from *G* to *H* is continuous.^{ [13] } Secondly, there is a version of the open mapping theorem or the closed graph theorem due to Kuratowski (1933 , p. 400) : a continuous injective homomorphism of a Polish subgroup *G* onto another Polish group *H* is an open mapping. As a result, it is a remarkable fact about Polish groups that Baire-measurable mappings (i.e., for which the preimage of any open set has the property of Baire) that are homomorphisms between them are automatically continuous.^{ [14] } The group of homeomorphisms of the Hilbert cube [0,1]^{N} is a universal Polish group, in the sense that every Polish group is isomorphic to a closed subgroup of it.

Examples:

- All finite dimensional Lie groups with a countable number of components are Polish groups.
- The unitary group of a separable Hilbert space (with the strong operator topology) is a Polish group.
- The group of homeomorphisms of a compact metric space is a Polish group.
- The product of a countable number of Polish groups is a Polish group.
- The group of isometries of a separable complete metric space is a Polish group

- ↑ Bourbaki 1989 , p. 197
- ↑ Bourbaki 1989 , p. 197
- ↑ Srivastava 1998 , p. 55
- ↑ Rogers & Williams 1994 , p. 126
- ↑ Bourbaki 1989
- ↑ Schwartz 1973 , p. 94
- ↑ Schwartz 1973 , p. 102, Corollary 2 of Theorem 5.
- ↑ Schwartz 1973 , pp. 94, 102, Lemma 4 and Corollary 1 of Theorem 5.
- ↑ Schwartz 1973 , pp. 95, Lemma 6.
- ↑ Schwartz 1973 , p. 95, Corollary of Lemma 5.
- ↑ Bourbaki 1989 , pp. 197–199
- ↑ Moore 1976 , p. 8, Proposition 5
- ↑ Freudenthal 1936 , p. 54
- ↑ Pettis 1950.

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

In mathematics, more specifically in general topology, **compactness** is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space in various ways.

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.

In topology and related areas of mathematics, a **metrizable space** is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is . **Metrization theorems** are theorems that give sufficient conditions for a topological space to be metrizable.

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

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 **R** of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space, a metric space, a topological space, a measure space, or a linear continuum.

In mathematics, a **Baire space** is a topological space such that every intersection of a countable collection of open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem. The spaces are named in honor of René-Louis Baire who introduced the concept.

In mathematics, the **Hilbert cube**, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube.

In functional analysis and related areas of mathematics, **Fréchet spaces**, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically *not* Banach spaces.

In the mathematical field of topology, a **G _{δ} set** is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with

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 2^{ω} is called "the" Cantor space.

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 finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.

In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In topology, a **second-countable space**, also called a **completely separable space**, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the **second axiom of countability**. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers. This space, denoted by , is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by

In topology and related areas of mathematics, a subset *A* of a topological space *X* is called **dense** if every point *x* in *X* either belongs to *A* or is a limit point of *A*; that is, the closure of *A* constitutes the whole set *X*. Informally, for every point in *X*, the point is either in *A* or arbitrarily "close" to a member of *A* — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

In functional analysis and related areas of mathematics, a **metrizable** topological vector spaces (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

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