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 $S$ of sets, consider the Cartesian product $\textstyle X=\prod _{Y\in S}Y\,$ of all sets in the collection. The canonical projection corresponding to some $Y\in S$ is the function $p_{Y}:X\to Y$ that maps every element of the product to its $Y$ 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,

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

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

$S^{\mathbb {Z} }=\{x=(\ldots ,x_{-1},x_{0},x_{1},\ldots ):x_{k}\in S\;\forall k\in \mathbb {Z} \}.$ The natural topology on $S$ is the discrete topology. Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on $S^{\mathbb {Z} )$ are

$C_{t}[a]=\{x\in S^{\mathbb {Z} }:x_{t}=a\}.$ The intersections of a finite number of open cylinders are the cylinder sets

{\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 $V$ over a field K (such as the real or complex numbers), the cylinder sets may be defined as

$C_{A}[f_{1},\ldots ,f_{n}]=\{x\in V:(f_{1}(x),f_{2}(x),\ldots ,f_{n}(x))\in A\)$ where $A\subset K^{n)$ is a Borel set in $K^{n)$ , and each $f_{j)$ is a linear functional on $V$ ; that is, $f_{j}\in (V^{*})^{\otimes n)$ , the algebraic dual space to $V$ . When dealing with topological vector spaces, the definition is made instead for elements $f_{j}\in (V^{\prime })^{\otimes n)$ , the continuous dual space. That is, the functionals $f_{j)$ are taken to be continuous linear functionals.

Applications

Cylinder sets are often used to define a topology on sets that are subsets of $S^{\mathbb {Z} )$ 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 $S^{\mathbb {Z} )$ 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.