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.^[1] 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.