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Cylinder set

## Summary

In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra.

## General definition

Given a collection ${\displaystyle S}$ of sets, consider the Cartesian product ${\displaystyle \textstyle X=\prod _{Y\in S}Y\,}$ of all sets in the collection. The canonical projection corresponding to some ${\displaystyle Y\in S}$ is the function ${\displaystyle p_{Y}:X\to Y}$ that maps every element of the product to its ${\displaystyle Y}$ component. A cylinder set is a preimage of a canonical projection or finite intersection of such preimages. Explicitly, it is a set of the form,

${\displaystyle \bigcap _{i=1}^{n}p_{Y_{i}}^{-1}\left(A_{i}\right)=\left\{\left(x\right)\in X\mid x_{Y_{1}}\in A_{1},\dots ,x_{Y_{n}}\in A_{n}\right\}}$

for any choice of ${\displaystyle n}$, finite sequence of sets ${\displaystyle Y_{1},...Y_{n}\in S}$ and subsets ${\displaystyle A_{i}\subseteq Y_{i}}$ for ${\displaystyle 1\leq i\leq n}$. Here ${\displaystyle x_{Y}\in Y}$ denotes the ${\displaystyle Y}$ component of ${\displaystyle x\in X}$.

Then, when all sets in ${\displaystyle S}$ are topological spaces, the product topology is generated by cylinder sets corresponding to the components' open sets. That is cylinders of the form ${\displaystyle \bigcap _{i=1}^{n}p_{Y_{i}}^{-1}\left(U_{i}\right)}$ where for each ${\displaystyle i}$, ${\displaystyle U_{i}}$ is open in ${\displaystyle Y_{i}}$. In the same manner, in case of measurable spaces, the cylinder σ-algebra is the one which is generated by cylinder sets corresponding to the components' measurable sets.

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 the latter case, the resulting topology is the box topology; cylinder sets are never Hilbert cubes.

## Cylinder sets in products of discrete sets

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

${\displaystyle S^{\mathbb {Z} }=\{x=(\ldots ,x_{-1},x_{0},x_{1},\ldots ):x_{k}\in S\;\forall k\in \mathbb {Z} \}.}$

The natural topology on ${\displaystyle S}$ is the discrete topology. Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on ${\displaystyle S^{\mathbb {Z} }}$ are

${\displaystyle C_{t}[a]=\{x\in S^{\mathbb {Z} }:x_{t}=a\}.}$

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

{\displaystyle {\begin{aligned}C_{t}[a_{0},\ldots ,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}\}\end{aligned}}.}

Cylinder sets are clopen sets. 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.

## Definition for vector spaces

Given a finite or infinite-dimensional vector space ${\displaystyle V}$ over a field K (such as the real or complex numbers), the cylinder sets may be defined as

${\displaystyle C_{A}[f_{1},\ldots ,f_{n}]=\{x\in V:(f_{1}(x),f_{2}(x),\ldots ,f_{n}(x))\in A\}}$

where ${\displaystyle A\subset K^{n}}$ is a Borel set in ${\displaystyle K^{n}}$, and each ${\displaystyle f_{j}}$ is a linear functional on ${\displaystyle V}$; that is, ${\displaystyle f_{j}\in (V^{*})^{\otimes n}}$, the algebraic dual space to ${\displaystyle V}$. When dealing with topological vector spaces, the definition is made instead for elements ${\displaystyle f_{j}\in (V^{\prime })^{\otimes n}}$, the continuous dual space. That is, the functionals ${\displaystyle f_{j}}$ are taken to be continuous linear functionals.

## Applications

Cylinder sets are often used to define a topology on sets that are subsets of ${\displaystyle S^{\mathbb {Z} }}$ 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, using the Kolmogorov extension theorem; for example, the measure of a cylinder set of length m might be given by 1/m or by 1/2m.

Cylinder sets may be used to define a 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.

Since strings in ${\displaystyle S^{\mathbb {Z} }}$ 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. These types of measure spaces appear in the theory of dynamical systems and are called nonsingular odometers. A generalization of these systems is the Markov odometer.

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