en>David Eppstein: {{comp-sci-theory-stub}}

2013-10-04T03:50:27Z

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	The ~~author~~'~~s title~~ is ~~Andera and she believes it seems fairly great~~. ~~Ohio~~ is ~~where his home~~ is ~~and his family loves it~~. ~~One~~ of the ~~very best issues~~ in the ~~world~~ for ~~him is performing ballet and he~~'~~ll be beginning~~ [~~http~~://~~www~~.~~weddingwall~~.~~com~~.au/~~groups~~/~~easy-advice-~~for-~~successful-personal-development-today~~/ real ~~psychic~~] ~~something else alongside~~ with ~~it. Distributing manufacturing has~~ [~~http:~~//~~help~~.~~ksu~~.~~edu~~.sa/~~node~~/~~65129 real psychic readings] been his occupation for some time~~.<br><br>~~Look into my blog~~ - ~~clairvoyance~~; [~~http://www~~.~~aseandate~~.~~com/index~~.~~php?m=member_profile&p~~=~~profile&~~id=~~13352970 just click the next document~~],		In [[mathematics]], a '''cylinder set''' is the natural [[Open_set#Topological_spaces\|open set]] of a [[product topology]]. Cylinder sets are particularly useful in providing the [[basis (topology)\|base]] of the [[natural topology]] of the product of a [[countable number]] of copies of a set. If ''V'' is a [[finite set]], then each element of ''V'' can be represented by a letter, and the countable product can be represented by the collection of [[sequence\|strings]] of letters.

			==General definition==
			Consider the [[cartesian product]] <math>\textstyle X = \prod_{\alpha} X_{\alpha}\,</math> of [[topological space]]s <math>X_\alpha</math>, indexed by some index <math>\alpha</math>. The '''canonical projection''' is the function <math>p_{\alpha} : X \to X_{\alpha}</math> that maps every element of the product to its <math>\alpha</math> component. Then, given any [[open set]] <math>U\subset X_\alpha</math>, the preimage <math>p_\alpha^{-1}(U)</math> is called an '''open cylinder'''. The intersection of a finite number of open cylinders is a '''cylinder set'''. The collection of open cylinders form a [[subbase]] of the [[product topology]] on <math>X</math>; the collection of all cylinder sets thus form a [[basis (topology)\|basis]].

			The restriction that the cylinder set be the intersection of a finite number of open cylinders is important; allowing infinite intersections generally results in a finer topology. In this case, the resulting topology is the [[box topology]]; cylinder sets are never [[Hilbert cube]]s.

			==Definition for infinite products of finite, discrete sets==
			Let <math>S=\{1,2,\ldots,n\}</math> be a [[finite set]], containing ''n'' objects or '''letters'''. The collection of all [[bi-infinite string]]s in these letters is denoted by

			:<math>S^\mathbb{Z}=\{ x=(\ldots,x_{-1},x_0,x_1,\ldots) :
			x_k \in S \; \forall k \in \mathbb{Z} \}</math>

			where <math>\mathbb{Z}</math> denotes the [[integer]]s. The natural topology on <math>S</math> is the [[discrete topology]]. Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on <math>S^\mathbb{Z}</math> are

			:<math>C_t[a]= \{x \in S^\mathbb{Z} : x_t = a \}.</math>

			The intersections of a finite number of open cylinders are the '''cylinder sets'''

			:<math>C_t[a_0, \cdots, a_m]= C_t[a_0] \,\cap\, C_{t+1}[a_1] \,\cap \cdots \cap\, C_{t+m}[a_m] =
			\{x \in S^\mathbb{Z} : x_t = a_0, \ldots ,x_{t+m} = a_m \}.</math>

			Cylinder sets are [[clopen set]]s. As elements of the topology, cylinder sets are by definition open sets. The complement of an open set is a closed set, but the complement of a cylinder set is a union of cylinders, and so cylinder sets are also closed, and are thus clopen. As a result, the topology satisfies the axioms of a [[sigma algebra]].

			==Definition for vector spaces==
			Given a finite or infinite-dimensional vector space <math>V</math> over a field ''K'' (such as the real or complex numbers), the cylinder sets may be defined as

			:<math>C_A[f_0, \cdots, f_m] = \{x\in V : (f_1(x),f_2(x),\cdots,f_m(x))\in A \}</math>

			where <math>A\subset K^n</math> is a [[Borel set]] in <math>K^n</math>, and each <math>f_j</math> is a linear functional on <math>V</math>; that is, <math>f_j\in (V^*)^{\otimes n}</math>, the algebraic [[dual space]] to <math>V</math>. When dealing with [[topological vector space]]s, the definition is made instead for elements <math>f_j\in (V^\prime)^{\otimes n}</math>, the continuous dual space. That is, the functionals <math>f_j</math> are taken to be continuous linear functionals. The article on [[dual space]]s discusses the differences between the algebraic and the continuous dual spaces.

			==Applications==
			Cylinder sets are often used to define a topology on sets that are subsets of <math>S^\mathbb{Z}</math> and occur frequently in the study of [[symbolic dynamics]]; see, for example, [[subshift of finite type]]. Cylinder sets are often used to define a [[measure theory\|measure]]; for example, the measure of a cylinder set of length ''m'' might be given by 1/''m'' or by <math>1/2^m</math>. Since strings in <math>S^\mathbb{Z}</math> can be considered to be [[p-adic number]]s, some of the theory of p-adic numbers can be applied to cylinder sets, and in particular, the definition of [[p-adic measure]]s and [[p-adic metric]]s apply to cylinder sets. Cylinder sets may be used to define a [[metric (mathematics)\|metric]] on the space: for example, one says that two strings are '''ε-close''' if a fraction 1-ε of the letters in the strings match.

			Cylinder sets over topological vector spaces are the core ingredient in the formal definition of the [[Feynman path integral]] or [[functional integral]] of [[quantum field theory]], and the [[partition function (mathematics)\|partition function]] of [[statistical mechanics]].

			==See also==
			* [[Cylinder set measure]]
			* [[Ultraproduct]]

			==References==
			* {{springer\|author=R.A. Minlos\|title=Cylinder Set\|id=C/c027620}}

			[[Category:General topology]]

en>Cydebot: Robot - Moving category Recursion theory to Computability theory per CFD at Wikipedia:Categories for discussion/Log/2011 February 5.

2011-02-14T19:59:02Z

Robot - Moving category Recursion theory to Computability theory per CFD at Wikipedia:Categories for discussion/Log/2011 February 5.

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en>David Eppstein: {{comp-sci-theory-stub}}

en>Cydebot: Robot - Moving category Recursion theory to Computability theory per CFD at Wikipedia:Categories for discussion/Log/2011 February 5.