Editing Base (topology) (section)

{{short description|Collection of open sets used to define a topology}}
In [[mathematics]], a '''base''' (or '''basis'''; {{plural form}}: '''bases''') for the [[Topology (structure)|topology]] {{math|τ}} of a [[topological space]] {{math|(''X'', τ)}} is a [[Family of sets|family]] <math>\mathcal{B}</math> of [[Open set|open subset]]s of {{math|''X''}} such that every open set of the topology is equal to the [[set union|union]] of some [[Subset|sub-family]] of <math>\mathcal{B}</math>.  For example, the set of all [[open interval]]s in the [[real number line]] <math>\R</math> is a basis for the [[Euclidean topology]] on <math>\R</math> because every open interval is an open set, and also every open subset of <math>\R</math> can be written as a union of some family of open intervals.

Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called {{em|basic open sets}}, are often easier to describe and use than arbitrary open sets.{{sfn|Adams|Franzosa|2009|pp=46-56}} Many important topological definitions such as [[Continuous function|continuity]] and [[Convergence (topology)|convergence]] can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.

Not all families of subsets of a set <math>X</math> form a base for a topology on <math>X</math>. Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on <math>X</math>, obtained by taking all possible unions of subfamilies.  Such families of sets are very frequently used to define topologies. A weaker notion related to bases is that of a [[subbase]] for a topology.  Bases for topologies are also closely related to [[neighborhood bases]].