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 primarily studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations.

Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and 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.

Read more about Polish Space: Properties, Characterization, Polish Metric Spaces

### Other articles related to "space, polish space, polish spaces, polish":

... The Baire

**space**has the following properties It is a perfect

**Polish space**, which means it is a completely metrizable second countable

**space**with no isolated points ... it has the same cardinality as the real line and is a Baire

**space**in the topological sense of the term ... It is universal for

**Polish spaces**in the sense that it can be mapped continuously onto any non-empty

**Polish space**...

... In the mathematical field of descriptive set theory, a subset of a

**Polish space**is projective if it is for some positive integer ... Here is if is analytic if the complement of, is if there is a

**Polish space**and a subset such that is the projection of that is, The choice of the

**Polish space**in the third clause above is ...

**Polish Space**s - 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 ...

... Every probability distribution on the

**space**turns it into a standard probability

**space**... on the Borel sigma-algebra and completed.) The same holds on every

**Polish space**, see (Rokhlin 1962, Sect ... For example, the Wiener measure turns the

**Polish space**(of all continuous functions endowed with the topology of local uniform convergence) into a standard probability

**space**...

### Famous quotes containing the words space and/or polish:

“This moment exhibits infinite *space*, but there is a *space* also wherein all moments are infinitely exhibited, and the everlasting duration of infinite *space* is another region and room of joys.”

—Thomas Traherne (1636–1674)

“Use the stones of another hill to *polish* your own jade.”

—Chinese proverb.