Base (topology)

In mathematics, a base (or basis; pl.: bases) for the topology τ of a topological space (X, τ) is a family ${\displaystyle {\mathcal {B}}}$ of open subsets of X such that every open set of the topology is equal to the union of some sub-family of ${\displaystyle {\mathcal {B}}}$. For example, the set of all open intervals in the real number line ${\displaystyle \mathbb {R} }$ is a basis for the Euclidean topology on ${\displaystyle \mathbb {R} }$ because every open interval is an open set, and also every open subset of ${\displaystyle \mathbb {R} }$ 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 basic open sets, are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and 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 ${\displaystyle X}$ form a base for a topology on ${\displaystyle X}$. Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on ${\displaystyle X}$, 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.