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Concept
Polish space
Summary
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.
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 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 Gδ-subset of X.
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.)
(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 Gδ-subset of the Hilbert cube (that is, of I^N, where I is the unit interval and N is the set of natural numbers).
Official source
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