What is a separable metric space

encyclopedia

In the mathematical discipline of general topology a Polish area is a separable, fully measurable topological space; That is, a space that is 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 studied today mainly because they represent the primary environment for descriptive set theory, including the study of Borel equivalence relationships. Polish spaces are also a convenient environment for more advanced measure theory, especially probability theory.

Common examples of Polish rooms are the real line, any separable Banach room, Cantor room, and Baire room. In addition, some spaces that are not full metric spaces in the common metric may be Polish. eg the open interval (0, 1) is Polish.

There is a Borel isomorphism between two innumerable Polish spaces; that is, a bijection that preserves the Borel structure. In particular, every innumerable Polish space has the cardinality of the continuum.

Lusin rooms, Suslin rooms, and Radon rooms are generalizations of Polish spaces.

Properties [edit]

  1. Every Polish area can be counted in the second place (since it can be measured separately).
  2. (Alexandrov's theorem) If X. is polish then everyone is like that Gδ Subset of X.. [1]
  3. A subspace Q. of a Polish area P. is Polish if and only if Q. is the intersection of a sequence of open subsets of P.. (This is the reverse of Alexandrov's theorem.)[2]
  4. (Cantor-Bendixson theorem) If X. is Polish then every closed subset of X. can be written as the disjoint union of a perfect set and a countable set. Next when the Polish area X. is innumerable, it can be written as the disjoint union of a perfect set and a countable open set.
  5. Every Polish area is homeomorphic to a Gδ- Subgroup of the Hilbert cube (i.e. from I, Where I is the unit interval and ℕ is the set of natural numbers).[3]

The following spaces are Polish:

  • closed subsets of a Polish area,
  • open subsets of a Polish area,
  • Products and disjoint unions of countable families of Polish areas,
  • locally compact spaces that can be measured and counted in infinity,
  • countable intersections of Polish subspaces of a topological space by Hausdorff,
  • the set of irrational numbers with the topology induced by the standard real line topology.

Characterization [edit]

There are numerous characterizations that show when a second countable topological space can be measured, such as the Urysohn metrization theorem. The problem of determining whether a measurable space is fully measurable is more difficult. Topological spaces like the interval of open units (0,1) can receive both complete and incomplete metrics that generate their topology.

There is a characterization of fully separable metric spaces in relation to a game known as the strong choquet game. A separable metric space is fully measurable if and only if the second player has a winning strategy in this game.

A second characterization follows from Alexandrow's theorem. It says that a separable metric space is fully measurable if and only if it is one

Subset of its completion in the original metric.

Polish metric spaces [edit]

Although Polish spaces are measurable, they are not metric spaces in and of themselves; Each Polish area allows many complete metrics leading to the same topology, but none of them are singled out or distinguished. A Polish space with an excellent complete metric is called a Polish metric space. An alternative approach, equivalent to the one given here, is to first define “Polish metric space” as “fully separable metric space” and then define “Polish space” as the topological space that consists of a Polish metric space through The metric is forgotten.

Generalizations of Polish spaces [edit]

Lusin rooms [edit]

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

There are many ways to create Lusin rooms. Specifically:

  • Every Polish area is a Lusin[6]
  • A subspace of a Lusin space is Lusin if and only if it is a Borel set.[7]
  • Every countable union or intersection of Lusin subspaces of a Hausdorff space is Lusin.[8]
  • The product of a countable number of Lusin spaces is Lusin.[9]
  • The disjoint union of a countable number of Lusin spaces is Lusin.[10]

Suslin rooms [edit]

A Suslin room is the image of a Polish area under continuous mapping. So every Lusin room is Suslin. In a Polish space a subset is a Suslin space if and only if it is a Suslin set (a picture of the Suslin operation).[11]

The following are Suslin rooms:

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

They have the following properties:

  • Each Suslin room can be separated.

Radon rooms [edit]

A Radon room, named after Johann Radon, is a topological space on which every Borel probability is measured M. is regular inside. Since a probability measure is globally finite and therefore a locally finite measure, every probability measure in a Radon space is also a Radon measure. In particular, a separable full metric space (M., d) is a radon room.

Every Suslin room is radon.

Polish groups [edit]

A Polish group is a topological group G this is also a Polish space, in other words homeomorphic to a separable full metric space. There are several classic results from Banach, Freudenthal and Kuratowski on homomorphisms between Polish groups.[12] First, Banach's argument (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 off G to H. is continuous.[13] Second, there is a version of the open mapping theorem or the closed graph theorem by Kuratowski (1933, p. 400). Harvtxt error: no target: CITEREFKuratowski1933 (help): a continuous injective homomorphism of a Polish subgroup G to another Polish group H. is an open assignment. As a result, it is a remarkable fact in Polish groups that Baire-measurable maps (ie for which the model of an open set has the property of Baire) that are homomorphisms between them are automatically continuous. 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:

  • All finite-dimensional Lie groups with a countable number of components are Polish groups.
  • The unified group of a separable Hilbert space (with the strong operator topology) is a Polish group.
  • The group of homeomorphisms of a compact metric space is a Polish group.
  • The product of a countable number of Polish groups is a Polish group.
  • The group of isometries of a separable full metric space is a Polish group

See also [edit]

  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, consequence 2 of sentence 5.
  8. ^Schwartz 1973, pp. 94, 102, Lemma 4 and Corollary 1 of Theorem 5.
  9. ^Schwartz 1973, p. 95, Lemma 6.
  10. ^Schwartz 1973, p. 95, corollary from Lemma 5.
  11. ^Bourbaki 1989, pp. 197-199
  12. ^Moore 1976, p. 8, sentence 5
  13. ^Freudenthal 1936, p. 54

References [edit]

  • Banach, Stefan (1932). Théorie des opérations linéaires. Monograph Matematyczne (in French). Warsaw.
  • Bourbaki, Nicolas (1989). “IX. Use of Real Numbers in General Topology ”. Elements of Mathematics: General Topology, Part 2. Springer publishing house. 3540193723.
  • Freudenthal, Hans (1936). “Some theorems about topological groups”. Ann. from Math.37: 46–56.
  • Kuratowski, K. (1966). Topology Vol. I. Academic press. ISBN.
  • Moore, Calvin C. (1976). “Group extensions and cohomology for locally compact groups. III ”. Trans. Amer. Mathematics. Soc.221: 1–33.
  • Pettis, BJ (1950). “On the continuity and openness of homomorphisms in topological groups”. Ann. from Math.51: 293–308.
  • Rogers, LCG; Williams, David (1994). Diffusions, Markov processes and Martingales, Volume 1: Fundamentals, 2nd edition. John Wiley & Sons Ltd.
  • Schwartz, Laurent (1973). Radon measurements in any topological space and cylindrical measurements. Oxford University Press. ISBN.
  • Srivastava, Sashi Mohan (1998). A course on Borel sets. Diploma texts in mathematics. Springer publishing house. ISBN. Retrieved 2008-12-04.

Further reading [edit]