In mathematics, a Polish space is a separable completely metrisable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In topology and related areas of mathematics a topological space is called separable if it contains a countable dense subset; that is, a set with a countable number of elements whose closure is the entire space. ... A metrizable space is a topological space that is homeomorphic to a metric space. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This word should not be confused with homomorphism. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics the term countable set is used to describe the size of a set, e. ... In topology and related areas of mathematics a subset A of a topological space X is called dense (in X) if the only closed subset of X containing A is X itself. ...

Common examples of Polish spaces are the real line, the Cantor space, and Baire space. In mathematics, the real line is simply the set of real numbers. ... In mathematics, the term Cantor space is sometimes used to denote the topological abstraction of the classical Cantor set: A topological space is a Cantor space if it is homeomorphic to the Cantor set. ... In mathematics, the Baire space is the set of all infinite sequences of natural numbers. ...

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. In mathematics, an uncountable set is a set which is not countable. ... In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space X is a σ-algebra of subsets of X associated to the topology of X. In the mathematics literature, there are at least two inequivalent definitions of this σ-algebra: The minimal σ-algebra containing the open sets. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ...

Polish spaces are the primary setting for descriptive set theory, including the study of Borel equivalence relations. In mathematics, descriptive set theory is the study of certain classes of well-behaved sets of real numbers, e. ... In mathematics, a Borel equivalence relation on a Polish space X is an equivalence relation on X that is a Borel subset of X × X (in the product topology). ...

References

• Kechris, Alexander S. (1994). Classical Descriptive Set Theory, Springer-Verlag. ISBN 0-387-94374-9.

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory, North Holland. ISBN 0-444-70199-0.