Polish space

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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.

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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 (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. Every Polish space is second countable (by virtue of being separable metrizable).
  2. (Alexandrov's theorem) If X is Polish then so is any Gδ subset of X. [1]
  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.) [2]
  4. (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.
  5. Every Polish space is homeomorphic to a Gδ-subset of the Hilbert cube (that is, of I, where I is the unit interval and is the set of natural numbers). [3]

The following spaces are Polish:

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 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 topological space is a Lusin space if it is homeomorphic to a Borel subset of a compact metric space. [4] [5] Some stronger topology makes a Lusin into a Polish space.

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

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). [11]

The following are Suslin spaces:

They have the following properties:

Radon spaces

A Radon space, named after Johann Radon, is a topological space such that every Borel probability measure on M is inner regular. Since a probability measure is globally finite, and hence a locally finite measure, every probability measure on a Radon space is also a Radon measure. In particular a separable complete metric space (M, d) is a Radon space.

Every Suslin space is Radon.

Polish groups

A Polish group is a topological group G that is also a Polish space, in other words homeomorphic to a separable complete metric space. There are several classic results of Banach, Freudenthal and Kuratowski on homomorphisms between Polish groups. [12] Firstly, the argument of Banach (1932 , p. 23) applies mutatis mutandi to non-Abelian Polish groups: if G and H are separable metric spaces with G Polish, then any Borel homomorphism from G to H is continuous. [13] Secondly, there is a version of the open mapping theorem or the closed graph theorem due to Kuratowski (1933 , p. 400): a continuous injective homomorphism of a Polish subgroup G onto another Polish group H is an open mapping. As a result, it is a remarkable fact about Polish groups that Baire-measurable mappings (i.e., for which the preimage of any open set has the property of Baire) that are homomorphisms between them are automatically continuous. [14] The group of homeomorphisms of the Hilbert cube [0,1]N is a universal Polish group, in the sense that every Polish group is isomorphic to a closed subgroup of it.

Examples:

See also

Notes

  1. Bourbaki 1989 , p. 197
  2. Bourbaki 1989 , p. 197
  3. Srivastava 1998 , p. 55
  4. Rogers & Williams 1994 , p. 126
  5. Bourbaki 1989
  6. Schwartz 1973 , p. 94
  7. Schwartz 1973 , p. 102, Corollary 2 of Theorem 5.
  8. Schwartz 1973 , pp. 94, 102, Lemma 4 and Corollary 1 of Theorem 5.
  9. Schwartz 1973 , pp. 95, Lemma 6.
  10. Schwartz 1973 , p. 95, Corollary of Lemma 5.
  11. Bourbaki 1989 , pp. 197–199
  12. Moore 1976 , p. 8, Proposition 5
  13. Freudenthal 1936 , p. 54
  14. Pettis 1950.

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