In mathematics, a **cylinder set** is a set in the standard basis for the open sets of the product topology; they are also a generating family of the cylinder σ-algebra, which in the countable case is the product σ-algebra.

- General definition
- Cylinder sets in products of discrete sets
- Definition for vector spaces
- Applications
- See also
- References

Cylinder sets are particularly useful in providing the base of the natural topology of the product of a countable number of copies of a set. If *V* is a finite set, then each element of *V* can be represented by a letter, and the countable product can be represented by the collection of strings of letters.

Given a collection of sets, consider the Cartesian product 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 of a canonical projection or finite intersection of such preimages. Explicitly, it is a set of the form,

for any choice of , finite sequence of sets and subsets for . Here denotes the component of .

Then, when all sets in are topological spaces, 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. For a countable product, the cylinder σ-algebra is the product σ-algebra.^{ [1] }

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.

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. 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 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.

Given a finite or infinite-dimensional vector space over a field *K* (such as the real or complex numbers), the cylinder sets may be defined as

where is a Borel set in , and each is a linear functional on ; that is, , the algebraic dual space to . When dealing with topological vector spaces, the definition is made instead for elements , the continuous dual space. That is, the functionals are taken to be continuous linear functionals.

Cylinder sets are often used to define a topology on sets that are subsets of 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 1/2^{m}.

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 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 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, and the partition function of statistical mechanics.

In mathematics, one can often define a **direct product** of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions.

In mathematics, the **inverse limit** is a construction that allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category, and they are a special case of the concept of a limit in category theory.

In mathematics, a **normed vector space** or **normed space** is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:

- It is nonnegative, that is for every vector x, one has
- It is positive on nonzero vectors, that is,
- For every vector x, and every scalar one has
- The triangle inequality holds; that is, for every vectors x and y, one has

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In mathematics, there are usually many different ways to construct a **topological tensor product** of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products, but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.

In mathematics, an **operad** is concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others. They form a category-theoretic analog of universal algebra.

In mathematics, a positive measure *μ* defined on a *σ*-algebra Σ of subsets of a set *X* is called a finite measure if *μ*(*X*) is a finite real number, and a set *A* in Σ is of finite measure if *μ*(*A*) < ∞*.* The measure *μ* is called **σ-finite** if *X* is the countable union of measurable sets with finite measure. A set in a measure space is said to have ** σ-finite measure** if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.

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In abstract algebra, a **completion** is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions *R* on a space *X* concentrates on a **formal neighborhood** of a point of *X*: heuristically, this is a neighborhood so small that *all* Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in case *R* has a metric given by a non-Archimedean absolute value.

In probability theory, a **standard probability space**, also called **Lebesgue–Rokhlin probability space** or just **Lebesgue space** is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.

In mathematics, a **cardinal function** is a function that returns cardinal numbers.

- ↑ Gerald B Folland (2013).
*Real Analysis: Modern Techniques and Their Applications*. John Wiley & Sons. p. 23. ISBN 0471317160.

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

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