In mathematics, a cylinder set is the natural open set of a product topology. Cylinder sets are particularly useful in providing the 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 strings of letters.

Definition

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Let $V=\backslash \{1,2,\backslash ldots,n\backslash \}$ be a finite set, containing n objects or letters. The collection of all bi-infinite strings in these letters is denoted by

$V^\backslash mathbb\{Z\}=\backslash \{\; x=(\backslash ldots,x\_\{-1\},x\_0,x\_1,\backslash ldots)\; :$

x_k \in V \; \forall k \in \mathbb{Z} \}

where $\backslash mathbb\{Z\}$ denotes the integers. This collection $V^\backslash mathbb\{Z\}$ has a natural topology, the product topology. The basis of the topology are the cylinder sets

$C\_t\backslash ldots,\; a\_m=\; \backslash \{x\; \backslash in\; V^\backslash mathbb\{Z\}\; :$

x_t = a_0, \ldots ,x_{t+m} = a_m \}

Cylinder sets are clopen sets.

Example

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The n-dimensional cylinder sets of C^* are defined by

$$

\mathcal{C}_n= \{ x \in C^* : x_{t_1} \in A_1, \ldots x_{t_n} \in A_n \}

where the $A\_i$ for i from 1 to n are Borel subsets of R. The cylinder sets of C^* are then defined by the union

$\backslash bar\; \backslash mathcal\{C\}\; =\; \backslash bigcup\_1^\backslash infty\; \backslash mathcal\{C\}\_n$

The σ algebra generated by the cylinder sets is defined to be the intersection of all σ algebras over Cwhich contains $\backslash bar\; \backslash mathcal\{C\}$. This σ algebra $\backslash mathcal\{C\}$ is frequently considered as the σ algebra of C[0,1 functions and is important in the development of the theory of continuous stochastic processes.

Applications

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Cylinder sets are often used to define a topology on sets that are subsets of $V^\backslash mathbb\{Z\}$ 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; for example, the measure of a cylinder set of length m might be given by 1/m or by $1/2^m$. Since strings in $V^\backslash mathbb\{Z\}$ can be considered to be p-adic numbers, some of the theory of p-adic numbers can be applied to cylinder sets, and in particular, the definition of p-adic measures and p-adic metrics apply to cylinder sets. Cylinder sets may be used to define a metric on the space: for example, one says that two strings are ε-close if 1/ε of the letters in the strings match.

See also

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• Box topology

General topology