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.
Other articles related to "polish space, polish, space, polish spaces":
... 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 ...
... 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 ...
... 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 ... 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 ...
... Every probability distribution on the space turns it into a standard probability space ... 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 ...
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