Borel hierarchy
In mathematics, and in particular the descriptive set theory, the Borel hierarchy is a gradual breakdown of the Borel σ - algebra on a topological space. It provides a structural design of all Borel sets dar. is a property of all Borel sets to prove this is often possible with transfinite induction on all levels of the Borel hierarchy.
Definition
About a topological space (amount of open sets ) the following quantity systems are defined inductively:
- .
- For each countable ordinal.
- For each countable ordinal.
- For each countable ordinal.
So called the open sets, the complements of sets, denotes the quantities that can be represented as a countable union of the sets for, and the amounts that are both in and in.
Completion and monotony properties
- The sets are closed under countable union and finite intersection.
- The sets are closed under countable intersection and finite union.
- The sets are closed under finite intersection, finite union and complementation.
- The quantities (as well as the quantities and the quantities ) are closed under continuous preimages, ie:
For a continuous function between topological spaces and, in turn, is a quantity ( amount, quantity), if an amount of ( amount, quantity) is.
- For all countable ordinals. In uncountable Polish spaces (which always have the cardinality of ) these inclusions are always strictly while already contains countable Polish space all subsets of the space.
Relation to the Borel σ - algebra
The union of all set systems of the Borel hierarchy is exactly the Borel σ - algebra, d i is the smallest σ - algebra containing all the open sets of the topological space.
That each amount in the Borel hierarchy in the Borel σ - algebra must be included immediately follows from the closure properties of a σ - algebra: If there were quantities in the Borel hierarchy, which are not included in the Borel σ - algebra, there would be a least ordinal so that such a contains (for the ordinals are well ordered ), which is equivalent to the fact that such has, since σ - algebras are closed under complementation. This element is, however, union of countably many elements of which are all Borel sets. Thus, the element should also be included in the σ - algebra, since σ - algebras are closed under countable union.
Conversely, all open sets contained in the Borel hierarchy and the amounts of the Borel hierarchy are closed under complementation and countable union: former follows directly from the definition of as complements, the second lets be shown as follows: Let a countable number of Borel - hierarchy given quantities occurring. For each there is an ordinal, so that, finally enter the quantities in the hierarchy. Then for the supremum, and the supremum of a set of ordinals is their union, therefore is countable union of countable sets and thus again a countable ordinal. Now it becomes clear why exactly countable ordinals are elected.
Relation to the projective hierarchy
Polish on the projective space hierarchy is defined starting from the Borel - hierarchy, which is projected from the Borel- volumes based on the analytical amounts. After Suslin theorem, the Borel sets in a Polish space exactly the quantities that are analytically and its complement is also analytically.
Dual definition on closed sets
The Borel hierarchy can also be defined on the basis of the closed sets:
- Be the set of all closed sets.
- For each countable ordinal.
- For each countable ordinal.
The quantities are thus defined as a countable intersection of sets for.