Topological defect: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Xiaoshan Math
No edit summary
en>Omnipaedista
per MOS:AMP
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
In [[mathematics]], a '''Cantor space''', named for [[Georg Cantor]], is a [[topology|topological]] abstraction of the classical [[Cantor set]]: a [[topological space]] is a '''Cantor space''' if it is [[homeomorphic]] to the [[Cantor set]]. In [[set theory]], the topological space 2<sup>ω</sup> is called "the" Cantor space.  Note that, commonly, 2<sup>ω</sup> is referred to simply as the Cantor set, while the term Cantor space is reserved for the more general construction of ''D''<sup>S</sup> for a finite set ''D'' and a set ''S'' which might be finite, countable or possibly uncountable.<ref>Stephen Willard, ''General Topology'' (1970) Addison-Wesley Publishing. ''See section 17.9a''</ref>


== Examples ==


The Cantor set itself is a Cantor space.  But the canonical example of a Cantor space is the [[countably infinite]] [[product topology|topological product]] of the [[discrete 2-point space]] {0, 1}. This is usually written as <math>2^\mathbb{N}</math> or 2<sup>ω</sup> (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2<sup>ω</sup> is an infinite binary sequence, that is a sequence which assumes only the values 0 or 1. Given such a sequence ''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>,..., one can map it to the real number
On top of the Clash of Clans hack tool; there are unquestionably also hack tools to suit other games. Men or women can check out all of the hacks and obtain hundreds of which they need. It is sure that will have lost of fun once they keep the hack tool that they can.<br><br>Products and solutions are a parent in addition gaurdian of any tad one who appreciates selecting part in video games, be familiar with multi-player and on-line video sports options. These properties give your kid in order to really interact with many some gamers throughout the world. As good as they is for your tyke in order to explain with others, you do not know who anyone on the other quit is.<br><br>Nevertheless, if you want in order to at the top of one's competitors, there are a few simple points you be obliged to keep in mind. Realize your foe, know which game and the success will be yours. It is possible consider the aid of clash of clans hack tools and other rights if you just like your course. So for your convenience, listed here the general details in this particular sport that you should try to [http://Www.britannica.com/search?query=remember remember] of. Read all of them scrupulously!<br><br>Program game playing is excellent kids. Consoles will offer you far better control of content and safety, all of the kids can simply breeze by way of parent regulates on your pc workstation. Using this step might help to shield your young ones from harm.<br><br>Always your game just as regularly as possible. While car-preservation is a good characteristic, do not count to sort it out. Particularly, when you in the start playing a game, you may not receive any thought when often the game saves, which could very well result in a drop of significant info as time goes on. Until you learn about the sport better, unfailingly save yourself.<br><br>You can see for yourself that this useful Money Compromise of Clans i fairly effective, really invisible by the manager of the game, substantially absolutely no price!<br><br>So as to master game play in shooter video games, guru your weapons. See everything there is learn about each and whatever weapon style in video game. Each weapon excels while in certain ways, but sheds short in others. When you know all the pluses and minuses relating to each weapon, you can easily use them to overly full advantage In the event you beloved this information along with you wish to get more info with regards to [http://prometeu.net triche clash of clans] kindly pay a visit to our own website. .
:<math>
\sum_{n=0}^\infty \frac{2 a_n}{3^{n+1}}.
</math>
This mapping gives a homeomorphism from 2<sup>ω</sup> onto the [[Cantor set]], demonstrating that 2<sup>ω</sup> is indeed a Cantor space.
 
Cantor spaces occur abundantly in [[real analysis]]. For example, they exist as subspaces in every perfect, [[complete space|complete]] [[metric space]].  (To see this, note that in such a space, any non-empty perfect set contains two disjoint non-empty perfect subsets of arbitrarily small diameter, and so one can imitate the construction
of the usual [[Cantor set]].)  Also, every uncountable,
[[separable space|separable]], completely metrizable space contains
Cantor spaces as subspaces.  This includes most of the common type of spaces in real analysis.
 
== Characterization ==
 
A topological characterization of Cantor spaces is given by [[Luitzen Egbertus Jan Brouwer|Brouwer]]'s theorem:{{citation needed|date=February 2014}}
:''Any two non-empty [[compact set|compact]] [[Hausdorff space]]s without [[isolated point]]s and having countable [[base (topology)|base]]s consisting of [[clopen set]]s are homeomorphic to each other''.
The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as: 
:''A topological space is a Cantor space if and only if it is non-empty, [[perfect space|perfect]], [[compact space|compact]], [[totally disconnected]], and [[metrizable]].''
This theorem is also equivalent (via [[Stone's representation theorem for Boolean algebras]]) to the fact that any two countable atomless [[Boolean algebra (structure)|Boolean algebra]]s are isomorphic.
 
== Properties ==
As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But many properties of Cantor spaces can be established using 2<sup>ω</sup>, because its construction as a product makes it amenable to analysis.  
 
Cantor spaces have the following properties:
* The [[cardinality]] of any Cantor space is <math>2^{\aleph_0}</math>, that is, the [[cardinality of the continuum]].
* The product of two (or even any finite or countable number of) Cantor spaces is a Cantor space. Along with the [[Cantor function]]; this fact can be used to construct [[space-filling curve]]s.
* A Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space.<ref>N.L. Carothers, ''A Short Course on Banach Space Theory'', London Mathematical Society Student Texts '''64''', (2005) Cambridge University Press. ''See Chapter 12''</ref><ref>Willard, ''op.cit.'', ''See section 30.7''</ref>
 
Let ''C(X)'' denote the space of all real-valued, bounded continuous functions on a topological space ''X''. Let ''K'' denote a compact metric space, and Δ denote the Cantor set. Then the Cantor set has the following property:
* C(K) is [[isometry|isometric]] to a closed subspace of C(Δ).<ref>Carothers, ''op.cit.''</ref>
In general, this isometry is not unique, and thus is not properly a [[universal property]] in the categorical sense.
 
*The group of all [[homeomorphisms]] of the Cantor space is [[simple group|simple]].<ref>R.D. Anderson, ''The Algebraic Simplicity of Certain Groups of Homeomorphisms'', American Journal of Mathematics '''80''' (1958), pp. 955-963.</ref>
 
==See also==
*[[Cantor cube]]
*[[Georg Cantor]]
 
==References==
<references/>
*{{cite book | author=Kechris, A. | title= Classical Descriptive Set Theory | publisher=Springer | year=1995 | isbn = 0-387-94374-9| edition=Graduate Texts in Mathematics 156}}
 
{{DEFAULTSORT:Cantor Space}}
[[Category:Topological spaces]]
[[Category:Descriptive set theory]]

Latest revision as of 19:22, 21 December 2014


On top of the Clash of Clans hack tool; there are unquestionably also hack tools to suit other games. Men or women can check out all of the hacks and obtain hundreds of which they need. It is sure that will have lost of fun once they keep the hack tool that they can.

Products and solutions are a parent in addition gaurdian of any tad one who appreciates selecting part in video games, be familiar with multi-player and on-line video sports options. These properties give your kid in order to really interact with many some gamers throughout the world. As good as they is for your tyke in order to explain with others, you do not know who anyone on the other quit is.

Nevertheless, if you want in order to at the top of one's competitors, there are a few simple points you be obliged to keep in mind. Realize your foe, know which game and the success will be yours. It is possible consider the aid of clash of clans hack tools and other rights if you just like your course. So for your convenience, listed here the general details in this particular sport that you should try to remember of. Read all of them scrupulously!

Program game playing is excellent kids. Consoles will offer you far better control of content and safety, all of the kids can simply breeze by way of parent regulates on your pc workstation. Using this step might help to shield your young ones from harm.

Always your game just as regularly as possible. While car-preservation is a good characteristic, do not count to sort it out. Particularly, when you in the start playing a game, you may not receive any thought when often the game saves, which could very well result in a drop of significant info as time goes on. Until you learn about the sport better, unfailingly save yourself.

You can see for yourself that this useful Money Compromise of Clans i fairly effective, really invisible by the manager of the game, substantially absolutely no price!

So as to master game play in shooter video games, guru your weapons. See everything there is learn about each and whatever weapon style in video game. Each weapon excels while in certain ways, but sheds short in others. When you know all the pluses and minuses relating to each weapon, you can easily use them to overly full advantage In the event you beloved this information along with you wish to get more info with regards to triche clash of clans kindly pay a visit to our own website. .

Retrieved from "https://en.formulasearchengine.com/index.php?title=Topological_defect&oldid=228970"