Cylinder set

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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.

1 General definition
2 Definition for infinite products of finite, discrete sets
3 Definition for vector spaces
4 Applications
5 See also
6 References

[edit] General definition

Consider the cartesian product

X =	∏	X_α
	α

of topological spaces $X α$ , indexed by some index $α$ . The canonical projection is the function $p_\alpha:X\to X_\alpha$ that selects out the $α$ component of the product. 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.

[edit] Definition for infinite products of finite, discrete sets

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

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

where $\mathbb{Z}$ denotes the integers. The natural topology on $S$ is the discrete topology. 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 open cylinders are the cylinder sets

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

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 again a cylinder set, and so cylinder sets are also closed, and are thus clopen. As a result, the topology satisfies the axioms of a sigma algebra.

[edit] 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.

[edit] 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 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 $S^\mathbb{Z}$ 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 1/ε of the letters in the strings match.