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]
(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.
Every Polish space is homeomorphic to a Gδ-subset of the Hilbert cube (that is, of IN, 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 space,
open subsets of a Polish space,
products and disjoint unions of countable families of Polish spaces,
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:
A subspace of a Lusin space is Lusin if and only if it is a Borel set.[7]
Any 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 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:
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,
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 mutandis 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) harvtxt error: no target: CITEREFKuratowski1933 (help): 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:
All finite dimensional Lie groups with a countable number of components are Polish groups.
Banach, Stefan (1932). Théorie des opérations linéaires. Monografie Matematyczne (in French). Warsaw.
Bourbaki, Nicolas (1989). "IX. Use of Real Numbers in General Topology". Elements of Mathematics: General Topology, Part 2. Springer-Verlag. 3540193723.
Pettis, B. J. (1950). "On continuity and openness of homomorphisms in topological groups". Ann. of Math.51 (2): 293–308. doi:10.2307/1969471. JSTOR 1969471.
Rogers, L. C. G.; Williams, David (1994). Diffusions, Markov Processes, and Martingales, Volume 1: Foundations, 2nd Edition. John Wiley & Sons Ltd.
Schwartz, Laurent (1973). Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press. ISBN978-0195605167.
Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel: ETH Zürich, Birkhäuser Verlag. ISBN3-7643-2428-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
polish, space, mathematical, discipline, general, topology, separable, completely, metrizable, topological, space, that, space, homeomorphic, complete, metric, space, that, countable, dense, subset, named, because, they, were, first, extensively, studied, poli. 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 Sierpinski 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 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 Contents 1 Properties 2 Characterization 3 Polish metric spaces 4 Generalizations of Polish spaces 4 1 Lusin spaces 4 2 Suslin spaces 4 3 Radon spaces 4 4 Polish groups 5 See also 6 References 7 Further readingProperties EditEvery Polish space is second countable by virtue of being separable metrizable Alexandrov s theorem If X is Polish then so is any Gd subset of X 1 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 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 Every Polish space is homeomorphic to a Gd subset of the Hilbert cube that is of IN 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 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 irrational numbers with the topology induced by the standard topology of the real line Characterization EditThere 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 d displaystyle G delta subset of its completion in the original metric Polish metric spaces EditAlthough 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 EditLusin spaces Edit 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 Every Polish space is Lusin 6 A subspace of a Lusin space is Lusin if and only if it is a Borel set 7 Any 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 spaces Edit 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 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 Edit 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 Edit 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 mutandis 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 harvtxt error no target CITEREFKuratowski1933 help 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 All finite dimensional Lie groups with a countable number of components are Polish groups The unitary 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 complete metric space is a Polish groupSee also EditStandard Borel spaceReferences Edit Bourbaki 1989 p 197 Bourbaki 1989 p 197 Srivastava 1998 p 55 Rogers amp Williams 1994 p 126 Bourbaki 1989 Schwartz 1973 p 94 Schwartz 1973 p 102 Corollary 2 of Theorem 5 Schwartz 1973 pp 94 102 Lemma 4 and Corollary 1 of Theorem 5 Schwartz 1973 pp 95 Lemma 6 Schwartz 1973 p 95 Corollary of Lemma 5 Bourbaki 1989 pp 197 199 Moore 1976 p 8 Proposition 5 Freudenthal 1936 p 54 Pettis 1950 Banach Stefan 1932 Theorie des operations lineaires Monografie Matematyczne in French Warsaw Bourbaki Nicolas 1989 IX Use of Real Numbers in General Topology Elements of Mathematics General Topology Part 2 Springer Verlag 3540193723 Freudenthal Hans 1936 Einige Satze ueber topologische Gruppen Ann of Math 37 1 46 56 doi 10 2307 1968686 JSTOR 1968686 Kuratowski K 1966 Topology Vol I Academic Press ISBN 012429202X Moore Calvin C 1976 Group extensions and cohomology for locally compact groups III Trans Amer Math Soc 221 1 33 doi 10 1090 S0002 9947 1976 0414775 X Pettis B J 1950 On continuity and openness of homomorphisms in topological groups Ann of Math 51 2 293 308 doi 10 2307 1969471 JSTOR 1969471 Rogers L C G Williams David 1994 Diffusions Markov Processes and Martingales Volume 1 Foundations 2nd Edition John Wiley amp Sons Ltd Schwartz Laurent 1973 Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures Oxford University Press ISBN 978 0195605167 Srivastava Sashi Mohan 1998 A Course on Borel Sets Graduate Texts in Mathematics Springer Verlag ISBN 978 0 387 98412 4 Retrieved 2008 12 04 Further reading EditAmbrosio L Gigli N amp Savare G 2005 Gradient Flows in Metric Spaces and in the Space of Probability Measures Basel ETH Zurich Birkhauser Verlag ISBN 3 7643 2428 7 a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link Arveson William 1981 An Invitation to C Algebras Graduate Texts in Mathematics Vol 39 New York Springer Verlag ISBN 0 387 90176 0 Kechris A 1995 Classical Descriptive Set Theory Graduate Texts in Mathematics Vol 156 Springer ISBN 0 387 94374 9 Retrieved from https en wikipedia org w index php title Polish space amp oldid 1099534785, wikipedia, wiki, book, books, library,
