, a subbase
) for a topological space X
with topology T
is a subcollection of T
such that every open set
can be written as a union of finite intersections of elements of B
. We say that the subbase generates
the topology T
, and that T
. The definition is used for two purposes in practice:
- First, if a topology T on a set X is already known, then a given subcollection of T can be singled out and shown to be a subbase for T. For instance, the usual topology on the real numbers R has a subbase consisting of all semi-infinite open intervals either of the form (−∞,a) or (a,∞), where a is a real number. Note that for any given topological space X with topology T, there may be more than one subbase for T. In the example above, the set of all semi-infinite open intervals either of the form (−∞,a) or (a,∞), where a is a rational number, is also a subbase for T. And clearly, for any topological space, the entire topology is itself a subbase. Thus, a subbase for a given fixed topology is not unique.
- Secondly, given a set X, a topology T on X can be specified by giving a subbase for T. In fact, for any subcollection S of the power set P(X) which coverss X, there is a unique smallest topology T containing S, and this topology T is called the topology generated by S. Compare this property to the behaviour of bases which generate topologies.