Base (topology)

In mathematics, a base (or basis; : bases) for the topology τ of a topological space (X, τ) is a family of open subsets of X such that every open set of the topology is equal to the union of some sub-family of . For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of 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 , 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 form a base for a topology on . Under some conditions detailed below, a family of subsets will form a base for a (unique) topology on , 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. Given a topological space , a base (or basis) for the topology (also called a base for if the topology is understood) is a family of open sets such that every open set of the topology can be represented as the union of some subfamily of . The elements of are called basic open sets. Equivalently, a family of subsets of is a base for the topology if and only if and for every open set in and point there is some basic open set such that . For example, the collection of all open intervals in the real line forms a base for the standard topology on the real numbers. More generally, in a metric space the collection of all open balls about points of forms a base for the topology. In general, a topological space can have many bases.