Cylinder set

Inmathematics,thecylinder setsform abasisof theproduct topologyon a product of sets; they are also a generating family of thecylinder σ-algebra.

General definition

Given a collection $S$ of sets, consider theCartesian product ${\textstyle X=\prod _{Y\in S}Y}$ of all sets in the collection. Thecanonical projectioncorresponding to some $Y\in S$ is thefunction $p_{Y}:X\to Y$ that maps every element of the product to its $Y$ component. A cylinder set is apreimageof a canonical projection or finiteintersectionof such preimages. Explicitly, it is a set of the form, $\bigcap _{i=1}^{n}p_{Y_{i}}^{-1}\left(A_{i}\right)=\left\{\left(x\right)\in X\mid p_{Y_{1}}(x)\in A_{1},\dots,p_{Y_{n}}(x)\in A_{n}\right\}$ for any choice of $n$ ,finite sequence of sets $Y_{1},...Y_{n}\in S$ andsubsets $A_{i}\subseteq Y_{i}$ for $1\leq i\leq n$ .

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

The restriction that the cylinder set be the intersection of afinitenumber of open cylinders is important; allowing infinite intersections generally results in afinertopology. In the latter case, the resulting topology is thebox topology;cylinder sets are neverHilbert cubes.

Cylinder sets in products of discrete sets

Let $S=\{1,2,\ldots,n\}$ be a finite set, containingnobjects orletters.The collection of allbi-infinite stringsin 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 thediscrete 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 thecylinder 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 areclopen 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 aunionof 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 afieldK(such as therealorcomplex 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 aBorel setin $K^{n}$ ,and each $f_{j}$ is alinear functionalon $V$ ;that is, $f_{j}\in (V^{*})^{\otimes n}$ ,thealgebraic dual spaceto $V$ .When dealing withtopological vector spaces,the definition is made instead for elements $f_{j}\in (V')^{\otimes n}$ ,thecontinuous 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 ofsymbolic dynamics;see, for example,subshift of finite type.Cylinder sets are often used to define ameasure,using theKolmogorov extension theorem;for example, the measure of a cylinder set of lengthmmight be given by $1/ m$ or by $1/2 m$ .

Cylinder sets may be used to define ametricon the space: for example, one says that two strings areε-closeif a fraction 1−ε of the letters in the strings match.

Since strings in $S^{\mathbb {Z} }$ can be considered to bep-adic numbers,some of the theory ofp-adic numbers can be applied to cylinder sets, and in particular, the definition ofp-adic measuresandp-adic metricsapply to cylinder sets. These types of measure spaces appear in the theory ofdynamical systemsand are callednonsingular odometers.A generalization of these systems is theMarkov odometer.

Cylinder sets over topological vector spaces are the core ingredient in the^{[citation needed]}definition ofabstract Wiener spaces,which provide the formal definition of theFeynman path integralorfunctional integralofquantum field theory,and thepartition functionofstatistical mechanics.

References

R.A. Minlos (2001) [1994],"Cylinder Set",Encyclopedia of Mathematics,EMS Press