Glossary of Topology - B

B

Baire space

This has two distinct common meanings:

1. A space is a Baire space if the intersection of any countable collection of dense open sets is dense; see Baire space.
2. Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see Baire space (set theory).

Base

A collection B of open sets is a base (or basis) for a topology if every open set in is a union of sets in . The topology is the smallest topology on containing and is said to be generated by .

Basis

See Base.

Borel algebra

The Borel algebra on a topological space is the smallest -algebra containing all the open sets. It is obtained by taking intersection of all -algebras on containing .

Borel set

A Borel set is an element of a Borel algebra.

Boundary

The boundary (or frontier) of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement. Boundary of a set is denoted by or .

Bounded

A set in a metric space is bounded if it has finite diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A function taking values in a metric space is bounded if its image is a bounded set.