Цилиндрлік жиынтар және топологиялық кеңістіктер

In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ algebra.

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
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. 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.

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 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, and the partition function of statistical mechanics.