Computable isomorphism

In mathematics, a cylinder set is the natural open set of a product topology. 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.

General definition

Consider the cartesian product $X=\prod _{\alpha }{X}_{\alpha }$ of topological spaces $X_{\alpha }$, indexed by some index $\alpha$. The canonical projection is the function $p_{\alpha }:X\to X_{\alpha }$ that maps every element of the product to its $\alpha$ component. Then, given any open set $U\subset X_{\alpha }$, the preimage $p_{\alpha }^{-1}(U)$ is called an open cylinder. The intersection of a finite number of open cylinders is a cylinder set. The collection of open cylinders form a subbase of the product topology on $X$; the collection of all cylinder sets thus form a basis.

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 this case, the resulting topology is the box topology; cylinder sets are never Hilbert cubes.

Definition for infinite products of finite, discrete sets

Let $S=\{1,2,\dots ,n\}$ be a finite set, containing n objects or letters. The collection of all bi-infinite strings in these letters is denoted by

$S^{\mathbb{\mathbb{Z}}}=\{x=(\dots ,x_{-1},x_0,x_1,\dots ):x_k\in S\mathrm{\forall }k\in \mathbb{\mathbb{Z}}\}$

where $\mathbb{\mathbb{Z}}$ denotes the integers. 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 $S^{\mathbb{\mathbb{Z}}}$ are

$C_t[a]=\{x\in S^{\mathbb{\mathbb{Z}}}:x_t=a\}.$

The intersections of a finite number of open cylinders are the cylinder sets

$C_t[a_0,\cdots ,a_m]=C_t[a_0]\cap C_{t+1}[a_1]\cap \cdots \cap C_{t+m}[a_m]=\{x\in S^{\mathbb{\mathbb{Z}}}:x_t=a_0,\dots ,x_{t+m}=a_m\}.$

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. As a result, the topology satisfies the axioms of a sigma algebra.

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

$C_A[f_0,\cdots ,f_m]=\{x\in V:(f_1(x),f_2(x),\cdots ,f_m(x))\in A\}$

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. The article on dual spaces discusses the differences between the algebraic and the continuous dual spaces.

Applications

Cylinder sets are often used to define a topology on sets that are subsets of $S^{\mathbb{\mathbb{Z}}}$ 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; for example, the measure of a cylinder set of length m might be given by 1/m or by $1/2^m$. Since strings in $S^{\mathbb{\mathbb{Z}}}$ 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. 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.

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.

See also

• Cylinder set measure
• Ultraproduct

References

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