Tata Institute of Fundamental Research: Difference between revisions

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In [[mathematics]], particularly in [[mathematical logic]] and [[set theory]], a '''club set''' is a subset of a [[limit ordinal]] which is [[closed set|closed]] under the [[order topology]], and is unbounded (see below) relative to the limit ordinal.  The name ''club'' is a contraction of "closed and unbounded".
== up three percent ==


== Formal definition ==
Spent [http://www.nrcil.net/fancybox/lib/rakuten_LV_99.html ルイヴィトン コレクション].<br><br>'how, how ......' Luo Feng incredible eyes [http://www.nrcil.net/fancybox/lib/rakuten_LV_67.html ルイヴィトン 六本木] wide [http://www.nrcil.net/fancybox/lib/rakuten_LV_73.html ルイヴィトン ベルニ] open.<br><br>how is it so hard to kill it?<br><br>sacrificed so much! How this Devourer is not dead! ! !<br><br>Golden Horn behemoth suspended [http://www.nrcil.net/fancybox/lib/rakuten_LV_71.html ルイヴィトン 柄] in mid-air, the breath of [http://www.nrcil.net/fancybox/lib/rakuten_LV_73.html ルイヴィトン ベルニ] life continued its strong, become strong ...... changed back to the original 10%, continue to [http://www.nrcil.net/fancybox/lib/rakuten_LV_22.html ルイヴィトン タイガ 財布] rise two percent, up three percent! That breath of life to rise to about three percent strength, there is no longer rising.<br><br>......<br>countless people on Earth<br>despair! Many people cry uncontrollably! Countless people suffering knees!<br><br>Gee!<br><br>......<br><br>Pacific Ocean waters.<br><br>Golden Horn monster eye slowly opened his eyes, dark golden eyes at the moment it is relatively dim weakness ...... it had never been so close [http://www.nrcil.net/fancybox/lib/rakuten_LV_76.html ルイヴィトン バック] to death! This time, even more than the last B6-level laser [http://www.nrcil.net/fancybox/lib/rakuten_LV_98.html ルイヴィトン 箱] cannon terrible.<br><br>'woo.' muffled roar powerless.<br><br>Golden Horn monster look to his abdomen.<br><br>saw
Formally, if <math>\kappa</math> is a limit ordinal, then a set <math>C\subseteq\kappa</math> is ''closed'' in <math>\kappa</math> [[if and only if]] for every <math>\alpha<\kappa</math>, if <math>\sup(C\cap \alpha)=\alpha\ne0</math>, then <math>\alpha\in C</math>. Thus, if the [[Limit of a sequence|limit of some sequence]] in <math>C</math> is less than <math>\kappa</math>, then the limit is also in <math>C</math>.
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If <math>\kappa</math> is a limit ordinal and <math>C\subseteq\kappa</math> then <math>C</math> is '''unbounded''' in <math>\kappa</math> if and only if for any <math>\alpha<\kappa</math>, there is some <math>\beta\in C</math> such that <math>\alpha<\beta</math>.
== ' Luo Feng always felt it was a time bomb. ==


If a set is both closed and unbounded, then it is a '''club set'''. Closed [[proper class]]es are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).  
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For example, the set of all [[countable]] limit ordinals is a club set with respect to the [[first uncountable ordinal]]; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.
<ul>
The set of all limit ordinals <math>\alpha<\kappa</math> is closed unbounded in <math>\kappa  </math>  (<math>\kappa  </math> regular).   In fact a  club set is nothing else but the range of   a [[normal function]]  (i.e. increasing and continuous).
 
 
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== The closed unbounded filter ==
 
Let <math>\kappa \,</math> be a limit ordinal of uncountable [[cofinality]] <math>\lambda \,.</math> For some <math>\alpha < \lambda \,</math>, let <math>\langle C_\xi : \xi < \alpha\rangle \,</math> be a sequence of closed unbounded subsets of <math>\kappa \,.</math> Then <math>\bigcap_{\xi < \alpha} C_\xi \,</math> is also closed unbounded. To see this, one can note that an intersection of closed sets is always closed, so we just need to show that this intersection is unbounded. So fix any <math>\beta_0 < \kappa \,,</math> and for each ''n''<&omega; choose from each <math>C_\xi \,</math> an element <math>\beta_{n+1}^\xi > \beta_{n} \,,</math> which is possible because each is unbounded. Since this is a collection of fewer than <math>\lambda \,</math> ordinals, all less than <math>\kappa \,,</math> their least upper bound must also be less than <math>\kappa \,,</math> so we can call it <math>\beta_{n+1} \,.</math> This process generates a countable sequence <math>\beta_0,\beta_1,\beta_2,\dots \,.</math> The limit of this sequence must in fact also be the limit of the sequence <math>\beta_0^\xi,\beta_1^\xi,\beta_2^\xi,\dots \,,</math> and since each <math>C_\xi \,</math> is closed and <math>\lambda \,</math> is uncountable, this limit must be in each <math>C_\xi \,,</math> and therefore this limit is an element of the intersection that is above <math>\beta_0 \,,</math> which shows that the intersection is unbounded. QED.
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From this, it can be seen that if <math>\kappa \,</math> is a regular cardinal, then <math>\{S \subset \kappa : \exists C \subset S \text{ such that } C \text{ is closed unbounded in } \kappa\} \,</math> is a non-principal <math>\kappa \,</math>-complete [[filter (mathematics)|filter]] on <math>\kappa \,.</math>
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If <math>\kappa \,</math> is a regular cardinal then club sets are also closed under [[diagonal intersection]].
</ul>
 
In fact, if <math>\kappa \,</math> is regular and <math>\mathcal{F} \,</math> is any filter on <math>\kappa \,,</math> closed under diagonal intersection, containing all sets of the form <math>\{\xi < \kappa : \xi \geq \alpha\} \,</math> for <math>\alpha < \kappa \,,</math> then <math>\mathcal{F} \,</math> must include all club sets.
 
==See also==
*[[Club filter]]
*[[Stationary set]]
*[[Clubsuit]]
 
==References==
* Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
* Levy, A. (1979) ''Basic Set Theory'', Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover. ISBN 0-486-42079-5
* {{PlanetMath attribution|id=3227|title=Club}}
 
[[Category:Set theory]]
[[Category:Ordinal numbers]]

Latest revision as of 13:11, 30 October 2014

up three percent

Spent ルイヴィトン コレクション.

'how, how ......' Luo Feng incredible eyes ルイヴィトン 六本木 wide ルイヴィトン ベルニ open.

how is it so hard to kill it?

sacrificed so much! How this Devourer is not dead! ! !

Golden Horn behemoth suspended ルイヴィトン 柄 in mid-air, the breath of ルイヴィトン ベルニ life continued its strong, become strong ...... changed back to the original 10%, continue to ルイヴィトン タイガ 財布 rise two percent, up three percent! That breath of life to rise to about three percent strength, there is no longer rising.

......
countless people on Earth
despair! Many people cry uncontrollably! Countless people suffering knees!

Gee!

......

Pacific Ocean waters.

Golden Horn monster eye slowly opened his eyes, dark golden eyes at the moment it is relatively dim weakness ...... it had never been so close ルイヴィトン バック to death! This time, even more than the last B6-level laser ルイヴィトン 箱 cannon terrible.

'woo.' muffled roar powerless.

Golden Horn monster look to his abdomen.

saw 相关的主题文章：

' Luo Feng always felt it was a time bomb.

Now there will not ルイヴィトンばっく be any threat to the Earth. 'Baba Ta explained,' Do not ルイヴィトン バッグ 人気 worry, you can ルイヴィトン バック now think of it as ...... 'sleep' state, and is permanently asleep. As for the details, so you will be 'the soul of India' practice to catch up with your ルイヴィトン タイガ 財布 teacher, you understand. '

'will not hurt, right?' Luo Feng always felt it was ルイヴィトン 財布 ダミエ a time bomb.

'No! unless the soul to a higher achievement than your teacher immortal.' Baba Ta laughed, ルイヴィトン 財布 定価 'that great presence, to destroy the Earth, Conspire to. would not say hard to get ルイヴィトン 六本木 That Zerg hive of living. '

Luo Feng That peace of mind, then wonder: '? teachers get this mother nest, ルイヴィトンさいふ do not have to kill it, in the end want to do.'

'Zerg Brood great potential, if you like, like slavery, servitude which Zerg hive, and then nurture, then your teacher will have an extremely frightening forces. endless Zerg army alone is enough In your teacher that death robbed 相关的主题文章：

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