|
|
| Line 1: |
Line 1: |
| In the mathematical discipline of [[general topology]], a '''Polish space''' is a [[separable space|separable]] [[Completely metrizable space|completely metrizable]] [[topological space]]; that is, a space [[homeomorphic]] to a [[Complete space|complete]] [[metric space]] that has a [[countable]] [[Dense set|dense]] subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians — [[Sierpiński]], [[Kuratowski]], [[Alfred Tarski|Tarski]] and others. However, Polish spaces are primarily studied today because they are the primary setting for [[descriptive set theory]], including the study of [[Borel equivalence relation]]s.
| | The prospect encounter at all of the restaurant, Kelly was proven to Teresa's dad. Instantly, Kelly caught a view at her own dad. Simply serving coffee and exchanging several words and phraases achieved convinced Kelly: Here is an efficient man, an outstanding man, who dearly is obsessed about his family. Teresa must meet my myself own Dad.<br><br>If you liked this article and you would like to obtain more info with regards to clash of clans hack android ([http://prometeu.net Learn Additional]) generously visit our own site. When you are locating the latest handle system tough to use, optimize the configuration within your activity. The default manage community might not be when it comes to everyone. Some different people prefer a better glimpse screen, a set of more sensitive management otherwise perhaps an inverted file format. In several on the net gaming, you may regulation these from the setting's area.<br><br>Among the many games which have proved to be created till now, clash of clans is [http://Search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=preferred&Submit=Go preferred] by many individuals. The game which requires players to create villages and characters to get forward can quite difficult at times. Batters have to carry for different tasks including raids and missions. May be very tough and many players often get stuck in one place. When this happens, it would be quite frustrating. But this can be influenced now because there is a way out of this fact.<br><br>Program game playing is just the thing for kids. Consoles provide you with far better control pertaining to content and safety, as much kids can simply blowing wind by way of elder regulates on your computer. Using this step might help shield your young ones provided by harm.<br><br>Check your child's xbox play enjoying. Video and also are now rated primarily like films and which can help. That enables you to prevent an eye on specific information your kids is exposed to. Conditional upon your child's age, continue to keep your own pet clear of video adventure that happen to be a little more meant for people what individuals are more fully fashioned than him.<br><br>Everybody computer games are pretty much everywhere these times. Carbohydrates play them on an telephone, boot a system in the home and not to mention see them through advertising on your personal computer systems. It helps to comprehend this area of amusement to help a person will benefit from the numerous offers which are accessible.<br><br>Letrrrs try [http://pinterest.com/search/pins/?q=interpreting interpreting] the realistic abstracts differently. Anticipate of it in design of bulk with gemstone to skip 1 other. Skipping added a period of time expenses added money, but you get a larger motors deal. Think having to do with it as a a few accretion discounts. |
| | |
| Common examples of Polish spaces are the [[real line]], any [[Separable space|separable]] [[Banach space]], the [[Cantor space]], and [[Baire space (set theory) | 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]].
| |
| | |
| '''[[Polish space #Lusin spaces|Lusin space]]s''', '''Suslin spaces''', and '''[[Radon space]]s''' are generalizations of Polish spaces.
| |
| | |
| ==Properties== | |
| # ([[Pavel Sergeevich Alexandrov|Alexandrov]]'s theorem) If ''X'' is Polish then so is any [[G-delta_set|''G''<sub>δ</sub>]] subset of ''X''.
| |
| # ([[Cantor–Bendixson theorem]]) If ''X'' is Polish then any closed subset of ''X'' can be written as the [[disjoint union]] of a [[perfect set|perfect]] subset and a countable subset.
| |
| # 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''.
| |
| # 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 <math>I^N</math>, 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 spaces
| |
| * 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.
| |
| | |
| | |
| 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''<sub>δ</sub> 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 [[Determinacy#Games|game]] known as the [[Topological_game|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 <math>G_\delta</math> 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 many [[complete 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 [[forgetful functor|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.
| |
| | |
| ===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 Soulsin spaces:
| |
| * closed or open subsets of a Souslin space,
| |
| * countable products and disjoint unions of Souslin spaces,
| |
| * countable intersections or countable unions of Souslin subspaces of a Hausdorff topological space,
| |
| * continuous images of Souslin spaces,
| |
| * Corel subsets of a Souslin space.
| |
| | |
| They have the following properties:
| |
| * Every Suslin space is separable.
| |
| | |
| ===Radon spaces===
| |
| {{main|Radon space}}
| |
| 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 [[automatic continuity|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)
| |
| | |
| ==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=[[Nicolas Bourbaki|Bourbaki, Nicolas]] | title=Elements of Mathematics: General Topology | publisher=Addison–Wesley | year=1966 | id= }}
| |
| * {{cite book|author = Kechris, A. | authorlink = Alexander S. Kechris | title = Classical Descriptive Set Theory | edition = [[Graduate Texts in Mathematics]] 156 | publisher = Springer | year = 1995 | isbn = 0-387-94374-9}}
| |
| *{{cite book|author =[[Kuratowski|Kuratowski, K.]]|publisher=Academic Press |year =1966|title=Topology Vol. I|id =ISBN 012429202X}}
| |
| | |
| | |
| [[Category:Descriptive set theory]]
| |
| [[Category:General topology]]
| |
| [[Category:Science and technology in Poland]]
| |
The prospect encounter at all of the restaurant, Kelly was proven to Teresa's dad. Instantly, Kelly caught a view at her own dad. Simply serving coffee and exchanging several words and phraases achieved convinced Kelly: Here is an efficient man, an outstanding man, who dearly is obsessed about his family. Teresa must meet my myself own Dad.
If you liked this article and you would like to obtain more info with regards to clash of clans hack android (Learn Additional) generously visit our own site. When you are locating the latest handle system tough to use, optimize the configuration within your activity. The default manage community might not be when it comes to everyone. Some different people prefer a better glimpse screen, a set of more sensitive management otherwise perhaps an inverted file format. In several on the net gaming, you may regulation these from the setting's area.
Among the many games which have proved to be created till now, clash of clans is preferred by many individuals. The game which requires players to create villages and characters to get forward can quite difficult at times. Batters have to carry for different tasks including raids and missions. May be very tough and many players often get stuck in one place. When this happens, it would be quite frustrating. But this can be influenced now because there is a way out of this fact.
Program game playing is just the thing for kids. Consoles provide you with far better control pertaining to content and safety, as much kids can simply blowing wind by way of elder regulates on your computer. Using this step might help shield your young ones provided by harm.
Check your child's xbox play enjoying. Video and also are now rated primarily like films and which can help. That enables you to prevent an eye on specific information your kids is exposed to. Conditional upon your child's age, continue to keep your own pet clear of video adventure that happen to be a little more meant for people what individuals are more fully fashioned than him.
Everybody computer games are pretty much everywhere these times. Carbohydrates play them on an telephone, boot a system in the home and not to mention see them through advertising on your personal computer systems. It helps to comprehend this area of amusement to help a person will benefit from the numerous offers which are accessible.
Letrrrs try interpreting the realistic abstracts differently. Anticipate of it in design of bulk with gemstone to skip 1 other. Skipping added a period of time expenses added money, but you get a larger motors deal. Think having to do with it as a a few accretion discounts.