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In mathematics, the cylinder sets form a basis of the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
on a product of sets; they are also a generating family of the
cylinder σ-algebra A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infi ...
.


General definition

Given a collection S of sets, consider the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...
X = \prod_ Y of all sets in the collection. The canonical projection corresponding to some Y\in S is the function p_ : X \to Y that maps every element of the product to its Y component. A cylinder set is a
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or throug ...
of a canonical projection or finite
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of such preimages. Explicitly, it is a set of the form, \bigcap_^n p_^ \left(A_i\right) = \left\ for any choice of n, finite sequence of sets Y_1,...Y_n\in S and
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s A_ \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 space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s, the product topology is generated by cylinder sets corresponding to the components' open sets. That is cylinders of the form \bigcap_^n p_^ \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 A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infi ...
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 cube In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, ...
s.


Cylinder sets in products of discrete sets

Let S = \ be a finite set, containing ''n'' objects or letters. The collection of all
bi-infinite string In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
s in these letters is denoted by S^\mathbb = \. The natural topology on S is the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest t ...
. Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on S^\mathbb are C_t \. The intersections of a finite number of open cylinders are the cylinder sets \begin C_t _0, \ldots, a_m& = C_t _0\,\cap\, C_ _1\,\cap \cdots \cap\, C_ _m\\ & = \ \end. Cylinder sets are
clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
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.


Definition for vector spaces

Given a finite or infinite- dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
V over a field ''K'' (such as the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...
or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s), the cylinder sets may be defined as C_A _1, \ldots, f_n= \ where A \subset K^n is a
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are name ...
in K^n, and each f_j is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , th ...
on V; that is, f_j\in (V^*)^, the
algebraic dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
to V. When dealing with
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...
s, the definition is made instead for elements f_j \in (V')^, the
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
. 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 and occur frequently in the study of
symbolic dynamics In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics ( ...
; see, for example,
subshift of finite type In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite state machin ...
. 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 or by . 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 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 In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
and are called
nonsingular odometer In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingu ...
s. A generalization of these systems is the Markov odometer. 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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
, and the partition function of statistical mechanics.


See also

* Cylindrical σ-algebra * Cylinder set measure *
Ultraproduct The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...


References

* {{springer, author=R.A. Minlos, title=Cylinder Set, id=C/c027620 General topology de:Zylindermenge