Polish space

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 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.

Properties

1. (Alexandrov's theorem) If X is Polish then so is any G_δ subset of X.
2. (Cantor–Bendixson theorem) If X is Polish then any closed subset of X can be written as the disjoint union of a perfect subset and a countable open subset.
3. 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.)
4. A topological space X is Polish if and only if X is homeomorphic to the intersection of a sequence of open subsets of the cube $I^N$, where I is the unit interval and N is the set of natural numbers.

The following spaces are Polish:

• closed subsets of a Polish space,
• open subsets of a Polish space,
• products and disjoint unions of countable families of Polish spaces,
• locally compact spaces that are metrizable and countable at infinity,
• countable intersections of Polish subspaces of a Hausdorff topological space,
• the set of nonrational numbers with the topology induced by the real line.

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 spaces; each Polish space admits many complete metrics 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 stronger topology makes it into a Polish space.

There are many ways to form Lusin spaces. In particular:

• Every Polish space is Lusin^[1]
• A subspace of a Lusin space is Lusin if and only if it is a Borel set.^[2]
• Any countable union or intersection of Lusin subspaces of a Hausdorff space is Lusin.^[3]
• The product of a countable number of Lusin spaces is Lusin.^[4]
• The disjoint union of a countable number of Lusin spaces is Lusin.^[5]

Suslin 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 a Suslin set (an image of the Suslin operation).

The following are Suslin spaces:

• closed or open subsets of a Suslin space,
• countable products and disjoint unions of Suslin spaces,
• countable intersections or countable unions of Suslin subspaces of a Hausdorff topological space,
• continuous images of Suslin spaces,
• Borel subsets of a Suslin space.

They have the following properties:

• Every Suslin space is separable.

Radon spaces

A Radon space is a topological space such that every finite Borel measure is inner regular (so a Radon measure). Every Suslin space is Radon.

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., the preimage of any open set has the property of Baire) homomorphisms between them are automatically continuous. (Pettis in B. J. Pettis, ‘On continuity and openness of homomorphisms in topological groups’, Ann. of Math. vol. 51 (1950) 293–308, MR 38358)

See also

• Standard Borel space

Notes

1. Schwartz, p. 94.
2. Schwartz, p. 102, Corollary 2 of Theorem 5.
3. Schwartz, p. 102, Corollary 1 of Theorem 5 and p. 94, Lemma 4.
4. Schwartz, p. 95, Lemma 6.
5. Schwartz, p. 95, Corollary of Lemma 5.

References

• Arveson, William (1981). An Invitation to C*-Algebras. Graduate Texts in Mathematics 39. New York: Springer-Verlag. ISBN 0-387-90176-0.
• Bourbaki, Nicolas (1966). Elements of Mathematics: General Topology. Addison–Wesley.
• Kechris, A. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics 156. Springer. ISBN 0-387-94374-9.
• Kuratowski, K. (1966). Topology Vol. I. Academic Press. ISBN 012429202X.
• Schwartz, Laurent (1973). Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press. ISBN 978-0195605167.