Editing Glossary of general topology (section)

== B ==

;Baire space: This has two distinct common meanings:
:#A space is a '''Baire space''' if the intersection of any [[countable]] collection of dense open sets is dense; see [[Baire space]].
:#'''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 (topology)|Base]]: A collection ''B'' of open sets is a [[base (topology)|base]] (or '''basis''') for a topology <math>\tau</math> if every open set in <math>\tau</math> is a union of sets in <math> B </math>. The topology <math>\tau</math> is the smallest topology on <math>X</math> containing <math>B</math> and is said to be generated by <math>B</math>.

;[[Basis (topology)|Basis]]: See '''[[Base (topology)|Base]]'''.

;β-open: See ''Semi-preopen''.

;b-open, b-closed: A subset <math>A</math> of a topological space <math>X</math> is b-open if <math>A \subseteq \operatorname{Int}_X \left( \operatorname{Cl}_X A \right) \cup \operatorname{Cl}_X \left( \operatorname{Int}_X A \right)</math>. The complement of a b-open set is b-closed.{{sfn|Hart|Nagata|Vaughan|2004|p=9}}

;[[Borel algebra]]: The [[Borel algebra]] on a topological space <math> (X,\tau)</math> is the smallest [[Sigma-algebra|<math>\sigma</math>-algebra]] containing all the open sets. It is obtained by taking intersection of all <math>\sigma</math>-algebras on <math> X </math> containing <math> \tau </math>.

;Borel set: A Borel set is an element of a Borel algebra.

;[[Boundary (topology)|Boundary]]: The [[boundary (topology)|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 <math> A </math> is denoted by <math> \partial A</math> or <math>bd</math> <math>A</math>.

;[[Bounded set|Bounded]]: A set in a metric space is [[bounded set|bounded]] if it has [[finite set|finite]] diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A [[function (mathematics)|function]] taking values in a metric space is [[bounded function|bounded]] if its [[image (functions)|image]] is a bounded set.