- Polish space
In

mathematics , a**Polish space**is a separable completely metrizabletopological space ; that is, a spacehomeomorphic to a completemetric space that has acountable 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 fordescriptive set theory , including the study ofBorel equivalence relation s.Common examples of Polish spaces are the

real line , theCantor space , and Baire space.Between any two

uncountable Polish spaces, there is a Borelisomorphism ; that is, abijection that preserves the Borel structure. In particular, every uncountable Polish space has thecardinality of the continuum .**Lusin spaces**,**Suslin spaces**, and**Radon spaces**are generalizations of Polish spaces.**Properties**# (Alexandrov's theorem) If "X" is Polish then so is any "G"

_{δ}subset of "X".

# (Cantor-Bendixson theorem) If "X" is Polish then any closed subset of "X" can be written as thedisjoint union of a perfect subset and a countable subset.The converse of Alexandrov's theorem is true as well: if a subspace "S" of a Polish space "X" is Polish, then it is a "G"

_{δ}subset of "X".**Characterization**There are numerous characterizations that tell when a second countable topological space is metrizable, such as

Urysohn's metrization theorem . The problem of determining whether a metrizable space is completely metrizable is more difficult. Topological spaces such as the open unit interval (0,1) can be given both complete metrics and incomplete metrics generating their topology.There is a characterization of complete separable metric spaces in terms of a game known as the strong Choquet game. A separable metric space is completely metrizable if and only if the second player has a

winning strategy in this game.A second characterization follows from Alexandrov's theorem. It states that a separable metric space is completely metrizable if and only if it is a $G\_delta$ subset of its completion in the original metric.

**Polish metric spaces**Although Polish spaces are metrizable, they are not in and of themselves

metric space s; each Polish space admits manycomplete metric s giving rise to the same topology, but no one of these is singled out or distinguished. A Polish space with a distinguished complete metric is called a**"Polish metric space**". An alternative approach, equivalent to the one given here, is first to define "Polish metric space" to mean "complete separable metric space", and then to define a "Polish space" as the topological space obtained from a Polish metric space by forgetting the metric.**Generalizations of Polish spaces****Lusin spaces**A

**Lusin space**is a topological space such that some weaker topology makes it into a Polish space.There are many ways to form Lusin spaces. In particular:

*Every Polish space is Lusin.

*A subspace of a Lusin space is Lusin if and only if it is a Borel set.

*Any countable union or intersection of Lusin subspaces of a Hausdorff space is Lusin.

*The product of a countable number of Lusin spaces is Lusin.

*The disjoint union of a countable number of Lusin spaces is Lusin.**uslin spaces**A

**Suslin space**is the image of a Polish space under a continuous mapping. So every Lusin space is Suslin.In a Polish space, a subset is a Suslin space if and only if it is aSuslin set (an image of the Suslin operation).**Radon spaces**A

**Radon space**is a topological space such that every finite Borel measure is inner regular (so aRadon measure ). Every Suslin space is Radon.**References***cite book | last=Arveson | first=William | authorlink=William Arveson | title=An Invitation to C*-Algebras | edition =

Graduate Texts in Mathematics 39 | publisher=Springer-Verlag | location=New York | year=1981 | isbn=0-387-90176-0

*cite book | author=Bourbaki, Nicolas | title=Elements of Mathematics: General Topology | publisher=Adison-Wesley | year=1966 | id=

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*cite book|author =Kuratowski, K.|publisher=Academic Press |year =1966|title=Topology Vol. I|id =ISBN 012429202X

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