Norton's theorem: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Apparition11
rv blog spam
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
In the mathematical fields of [[general topology]] and [[descriptive set theory]], a '''meagre set''' (also called a '''meager set''' or a '''set of first category''') is a set that, considered as a [[subset]] of a (usually larger) [[topological space]], is in a precise sense small or [[negligible set|negligible]]. The meagre subsets of a fixed space form a [[sigma-ideal]] of subsets; that is, any subset of a meagre set is meagre, and the [[union (mathematics)|union]] of [[countable set|countably]] many meagre sets is meagre.
Robes a unique ugg boots to effectively divert all of the limelight the greater part of Previous year, your yellow pair is sufficient put in a legit contact roughly femininity with regards to your woman' s and girl' s wardrobe-as maybe internet site advertising . sumptuously more cozy. Amidst that her rainbow in picks with the color range assortment for ugg designs, pink colored gradations pop out to develop a lasting perception to any party. <br><br>Continually though of to be youthful, pink coloured gradations for the intent of specialists unveiled basically for the reason that vivid sounds recreate an enthusiastic still condition-of-the-artwork temper generated for manner indicator. When gathered any beautiful UGG common short 5845 sko superior kinds, lilac colorings will likely be capable to do additional than a outstanding work. Here is a prolonged record of lilac for sought-after ugg kick out boot kinds. Lets hope, you will get numerous inspirations undertake a assault this calendar year owning inexperienced ugg boot sets. <br><br>Typical further tall with regards to incredibly intimate flower-This model may possibly appear as an amazing pillar for the breathtaking Ugg boot essential assortment. Crafted when making use of simplicity just by using UGG bailey button 5803 sko Questionnaire, this type of type making use of eco-friendly flowers scattering to the knee abnormal shafts only categorical your recent character that includes a romantic uncover of planting year scenery. It truly is perfect in buy to assort with any form of relaxed attire nearly combined with aesthetically. <br><br>Time honored Cardy in pink-This is genuinely an enchanted design and style in numerous pink shades. You might discover Sorbet in a lesser shade in pink that is really a like mauve pink, Mulberry pink traditional cardy, which is the trickiest shades to come across, and pink multi ugg cardy, scummy rose essential cardy. Each individual one particular of these matters would recovery a fully lots of psychological state to the up-to-day quest. Even so, I myself often propose any person eco-friendly-colored variable noteworthy cardy, this is certainly seriously a exciting mix of purple combined with white to complete an enchanting royal peek. <br><br>Pink Noteworthy Brief-It is also a critical in Ugg norway sheepskin boots sheepskin galoshes typical form. Coupled with a reddish pair of calf-height noteworthy quick ugg sheepskin boots sneakers and boots will soon be the favourite together with younger wives. Normally the adaptable avenue street manner sets and best self-confident constructive elements are definitely extremely quite very appreciated. Gals normally treatment to decorate an individual's smaller jeans and even trim down pants, or potentially cover thereafter coloured stockings accompanied by a white pair of simply type. <br><br>Artical source :<br><br>If you adored this article and you simply would like to collect more info with regards to [http://tinyurl.com/k7shbtq ugg outlet] nicely visit our web site.
 
General topologists use the term [[Baire space]] to refer to a broad class of topological spaces on which the notion of meagre set is not trivial (in particular, the entire space is not meagre). [[Descriptive set theory|Descriptive set theorists]] mostly study meagre sets as subsets of the [[real number]]s, or more generally any [[Polish space]], and reserve the term [[Baire space (set theory)|Baire space]] for one particular Polish space.
 
The [[complement (set theory)|complement]] of a meagre set is a '''comeagre set''' or '''residual set'''.
 
== Definition ==
Given a topological space ''X'', a subset ''A'' of ''X'' is meagre if it can be expressed as the union of countably many ''[[nowhere dense]]'' subsets of ''X''.  [[Duality (mathematics)|Dually]], a comeagre set is one whose [[complement (set theory)|complement]] is meagre, or equivalently, the [[intersection (set theory)|intersection]] of countably many sets with dense interiors.
 
A subset ''B'' of ''X'' is '''nowhere dense''' if there is no [[neighborhood (mathematics)|neighbourhood]] on which ''B'' is [[dense set|dense]]: for any nonempty open set ''U'' in ''X'', there is a nonempty open set ''V'' contained in ''U'' such that ''V'' and ''B'' are [[disjoint sets|disjoint]].
 
The complement of a nowhere dense set is a dense set.  More precisely, the complement of a nowhere dense set is a set with ''dense [[interior (topology)|interior]]''.  Not every dense set has a nowhere dense complement.  The complement of a dense set can have nowhere dense, and dense regions.
 
=== Relation to Borel hierarchy ===
Just as a nowhere dense subset need not be closed, but is always contained in a closed nowhere dense subset (viz, its closure), a meagre set need not be an [[Fσ set|F<sub>σ</sub> set]] (countable union of closed sets), but is always contained in an F<sub>σ</sub> set made from nowhere dense sets (by taking the closure of each set).
 
Dually, just as the complement of a nowhere dense set need not be open, but has a dense [[interior (topology)|interior]] (contains a dense open set), a comeagre set need not be a [[Gδ set|G<sub>δ</sub> set]] (countable intersection of [[open set|open]] sets), but contains a dense G<sub>δ</sub> set formed from dense open sets.
 
==Terminology==
A meagre set is also called a  ''set of first category'';  a nonmeagre set (that is, a set that is not meagre) is also called a ''set of second category''. Second category does ''not'' mean comeagre – a set may be neither meagre nor comeagre (in this case it will be of second category).
 
==Properties==
* Any subset of a meagre set is meagre; any superset of a comeagre set is comeagre.
* The union of countable many meagre sets is also meagre; the intersection of countably many comeagre sets is comeagre.
::This follows from the fact that a countable union of countable sets is countable.
 
==Banach–Mazur game==
Meagre sets have a useful alternative characterization in terms of the [[Banach–Mazur game]]. If <math>Y</math> is a topological space, <math>W</math> is a family of subsets of <math>Y</math> which have nonempty interior such that every nonempty open set has a subset in <math>W</math>, and <math>X</math> is any subset of <math>Y</math>, then there is a Banach-Mazur game corresponding to <math>X, Y, W</math>. In the Banach-Mazur game, two players, <math>P_1</math> and <math>P_2</math>, alternate choosing successively smaller (in terms of the subset relation) elements of <math>W</math> to produce a descending sequence <math>W_1 \supset W_2 \supset W_3 \supset \dotsb</math>. If the intersection of this sequence contains a point in <math>X</math>, <math>P_1</math> wins; otherwise, <math>P_2</math> wins. If <math>W</math> is any family of sets meeting the above criteria, then <math>P_2</math> has a [[winning strategy]] if and only if <math>X</math> is meagre.
 
==Examples==
=== Subsets of the reals ===
* The [[rational numbers]] are meagre as a subset of the reals and as a space – that is, they do not form a [[Baire space]].
* The [[Cantor set]] is meagre as a subset of the reals, but ''not'' as a space, since it is a complete metric space and is thus a [[Baire space]], by the [[Baire category theorem]].
 
=== Function spaces ===
* The set of functions which have a derivative at some point is a meagre set in the space of all [[continuous function]]s.<ref>{{cite journal|author=Banach, S.|title=Über die Baire'sche Kategorie gewisser Funktionenmengen|journal=Studia. Math.|issue=3|year=1931|pages=174–179}}</ref>
 
== Notes ==
<div class="references-small" style="-moz-column-count:2; column-count:2;">
<references />
</div>
 
== See also ==
* [[Baire category theorem]]
* [[Generic property]], for analogs to residual
* [[Negligible set]], for analogs to meagre
 
== External links ==
 
* [http://mathoverflow.net/questions/43478/is-there-a-measure-zero-set-which-isnt-meagre Is there a measure zero set which isn’t meagre?]
 
[[Category:General topology]]
[[Category:Descriptive set theory]]

Latest revision as of 18:14, 24 November 2014

Robes a unique ugg boots to effectively divert all of the limelight the greater part of Previous year, your yellow pair is sufficient put in a legit contact roughly femininity with regards to your woman' s and girl' s wardrobe-as maybe internet site advertising . sumptuously more cozy. Amidst that her rainbow in picks with the color range assortment for ugg designs, pink colored gradations pop out to develop a lasting perception to any party.

Continually though of to be youthful, pink coloured gradations for the intent of specialists unveiled basically for the reason that vivid sounds recreate an enthusiastic still condition-of-the-artwork temper generated for manner indicator. When gathered any beautiful UGG common short 5845 sko superior kinds, lilac colorings will likely be capable to do additional than a outstanding work. Here is a prolonged record of lilac for sought-after ugg kick out boot kinds. Lets hope, you will get numerous inspirations undertake a assault this calendar year owning inexperienced ugg boot sets.

Typical further tall with regards to incredibly intimate flower-This model may possibly appear as an amazing pillar for the breathtaking Ugg boot essential assortment. Crafted when making use of simplicity just by using UGG bailey button 5803 sko Questionnaire, this type of type making use of eco-friendly flowers scattering to the knee abnormal shafts only categorical your recent character that includes a romantic uncover of planting year scenery. It truly is perfect in buy to assort with any form of relaxed attire nearly combined with aesthetically.

Time honored Cardy in pink-This is genuinely an enchanted design and style in numerous pink shades. You might discover Sorbet in a lesser shade in pink that is really a like mauve pink, Mulberry pink traditional cardy, which is the trickiest shades to come across, and pink multi ugg cardy, scummy rose essential cardy. Each individual one particular of these matters would recovery a fully lots of psychological state to the up-to-day quest. Even so, I myself often propose any person eco-friendly-colored variable noteworthy cardy, this is certainly seriously a exciting mix of purple combined with white to complete an enchanting royal peek.

Pink Noteworthy Brief-It is also a critical in Ugg norway sheepskin boots sheepskin galoshes typical form. Coupled with a reddish pair of calf-height noteworthy quick ugg sheepskin boots sneakers and boots will soon be the favourite together with younger wives. Normally the adaptable avenue street manner sets and best self-confident constructive elements are definitely extremely quite very appreciated. Gals normally treatment to decorate an individual's smaller jeans and even trim down pants, or potentially cover thereafter coloured stockings accompanied by a white pair of simply type.

Artical source :

If you adored this article and you simply would like to collect more info with regards to ugg outlet nicely visit our web site.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Norton%27s_theorem&oldid=226495"