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Concept
Cylinder set
Summary
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 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.
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 .
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