- 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.