In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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-seemin ...
on a product of sets; they are also a generating family of the
cylinder σ-algebra.
General definition
Given a collection
of sets, consider the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of all sets in the collection. The canonical projection corresponding to some
is the
function that maps every element of the product to its
component. A cylinder set is a
preimage
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y.
More generally, evaluating f at each ...
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, their ...
of such preimages. Explicitly, it is a set of the form,
for any choice of
, finite sequence of sets
and
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
s
for
.
Then, when all sets in
are
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, the product topology is
generated by cylinder sets corresponding to the components' open sets. That is cylinders of the form
where for each
,
is open in
. 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 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, ca ...
s.
Cylinder sets in products of discrete sets
Let
be a finite set, containing ''n'' objects or letters. The collection of all
bi-infinite strings in these letters is denoted by
The natural topology on
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 to ...
. Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on
are
The intersections of a finite number of open cylinders are the cylinder sets
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 counterintuitive, as the common meanings of and are antonyms, but their mathematical de ...
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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
over a
field ''K'' (such as the
real 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
where
is a
Borel set
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
in
, and each
is a
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
on
; that is,
, the
algebraic dual space to
. 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 als ...
s, the definition is made instead for elements
, 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 const ...
. That is, the functionals
are taken to be continuous linear functionals.
Applications
Cylinder sets are often used to define a topology on sets that are subsets of
and occur frequently in the study of
symbolic dynamics
In mathematics, symbolic dynamics is the study of dynamical systems defined on a discrete space consisting of infinite sequences of abstract symbols. The evolution of the dynamical system is defined as a simple shift of the sequence.
Because of t ...
; 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 machi ...
. 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
Metric or metrical may refer to:
Measuring
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
...
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
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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
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 definition of
abstract Wiener spaces, which provide 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 Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, and the
partition function of
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
.
See also
*
*
*
*
*
*
References
*
{{Measure theory
General topology
Measure theory
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