In the mathematical discipline of
general topology, a Polish space is a
separable completely metrizable topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
; that is, a space
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to a
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
that has a
countable dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
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
In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" 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 applications to ot ...
, including the study of
Borel equivalence relation In mathematics, a Borel equivalence relation on a Polish space ''X'' is an equivalence relation on ''X'' that is a Borel algebra, Borel subset of ''X'' × ''X'' (in the product topology).
Formal definition
Given Borel equivalence relation ...
s. Polish spaces are also a convenient setting for more advanced
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
, in particular in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
.
Common examples of Polish spaces are the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, any
separable Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) 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 vector ...
, 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
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
(0, 1) is Polish.
Between any two
uncountable Polish spaces, there is a
Borel isomorphism; that is, a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
that preserves the Borel structure. In particular, every uncountable Polish space has the
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 \mathfrak c (lowercase fraktur "c") or , \mathb ...
.
Lusin spaces,
Suslin spaces, and
Radon spaces are generalizations of Polish spaces.
Properties
# Every Polish space is
second countable
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 T is second-countable if there exists some countable collection \mat ...
(by virtue of being separable metrizable).
# (
Alexandrov's theorem) If is Polish then so is any subset of .
# A subspace of a Polish space is Polish if and only if is the intersection of a sequence of open subsets of . (This is the converse to Alexandrov's theorem.)
# (
Cantor–Bendixson theorem 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 per ...
) If is Polish then any closed subset of can be written as the
disjoint union of a
perfect set
In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all Limit point, limit points of S, also known as the derived se ...
and a countable set. Further, if the Polish space 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 -subset of the
Hilbert cube
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, c ...
(that is, of , where is the unit interval and is the set of natural numbers).
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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
spaces that are metrizable and