Subbasis

A subbasis for a topology on a set X is a collection S of subsets of X whose presence determines a topology on X. Formally, S is a subbasis if the topology generated by S is the coarsest topology on X that contains S.

The topology generated by S, often denoted τ(S) or simply τ, consists of all unions of finite intersections of elements of S. In other words, take all finite products of members of S (where a product means intersection; at least one factor is used), and then form all possible unions of those intersections. The finite intersections of S elements form a basis for τ.

A few practical points. If S is empty, the generated topology is the indiscrete (trivial) topology containing only ∅ and X. A subbasis need not be closed under finite intersections; that property is not required because the topology is built from those finite intersections as a basis.

Relation to a basis. Every topology on X has at least one subbasis: the collection of all its open sets is a subbasis, and finite intersections of those opens form a basis for the topology. Conversely, any subbasis defines a unique topology—the smallest topology containing all subbasis elements.

Examples. In R with the standard topology, the collection of all open rays (-∞, a) and (a, ∞) for a ∈ R serves as a subbasis; finite intersections yield all open intervals, which generate the usual topology via unions. The empty subbasis yields the indiscrete topology, while choosing S to contain all singletons and their complements can generate various constructed topologies.

Subbases provide a flexible tool for defining topologies succinctly, especially when a direct basis is cumbersome to specify.