BorelAlgebra

Borelalgebra, often called the Borel algebra or Borel sigma-algebra, of a topological space X is the smallest sigma-algebra on X that contains every open set. Equivalently, it is the sigma-algebra generated by the open sets (or by the closed sets, since the complements of open sets are closed). This sigma-algebra includes all open sets, all closed sets, and any countable unions, intersections, and complements of such sets. It provides the natural domain for measurable functions and underpins the construction of Borel measures.

In practice, the Borel algebra on a concrete space such as the real numbers with the standard topology is generated by the open intervals (a,b) with real a < b. It contains all open and closed sets, as well as many more sets formed through countable operations, such as G_delta and F_sigma sets. Since topologies frequently have a countable basis (for instance, the open intervals with rational endpoints in R), the Borel algebra is often countably generated in such spaces.

Notes and variations: The term Borelalgebra is sometimes used interchangeably with Borel sigma-algebra. Some authors reserve the term “algebra” for a field closed under finite unions and complements, while “sigma-algebra” emphasizes closure under countable operations; in most mathematical contexts, Borel algebra refers to the sigma-algebra generated by the topology.

Applications: The Borel algebra is central in measure theory and probability, defining measurable maps, random variables, and probability measures on topological spaces. It also serves as a fundamental object in analysis and descriptive set theory.

See also: Borel sigma-algebra, measurable space, topology, sigma-algebra.