Borel set

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article is a stub and thus not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

In mathematics, a Borel set is a set that belongs to the σ-algebra generated by the open sets of a topological space. Thus, every open set is a Borel set, as are countable unions of open sets (i.e., unions of countably many open sets), and countable intersections of countable unions of open sets, etc.

Formal definition

Let be a topological space, i.e. is a set and are the open sets of (or, equivalently, the topology of ). Then is a Borel set of if , where denotes the σ-algebra generated by .

The σ-algebra generated by is simply the smallest σ-algebra containing the sets in or, equivalently, the intersection of all σ-algebras containing .