There is an alternate definition of the basis generated by a topology that is often useful:
Theorem 1 Let be a basis for a topology on . Then if and only if
Proof: Let . Then for each , there exists such that . It follows that . Conversely, if is the union of open sets , then for each , there is a such that , and so there is a basis element with , which implies that .
Bases are convenient for defining topologies whose members cannot easily be generically expressed. For example, given a metric space , the collection of open balls, that is, sets of the form
is a basis for topology on . For clearly
and if then and , so letting , we have
Indeed, the “usual topology” on , with a set being open iff for each there exists such that implies , is generated by this basis. If is open in , then
that is, open sets in are unions of open balls.
More generally, there is the notion of a subbasis:
Definition 2 A collection of subsets of is a subbasis for a topology on if . The topology generated by is the collection of sets of the form
In particular, given a family of topological spaces , we define the product topology as that generated by the subbasis
is the projection map defined by .
Theorem 3 The box topology is the same as the product topology when is finite and strictly finer than the product topology when is infinite.
Proof: If , then for any and open in with , we have
so is an element of the basis for the product topology.
If is infinite, then an element of the basis for the product topology is of the form
where is open in , so
If is open in for each , then
is an element of if and only if for all but finitely many .
For example, the box topology on is generated by the open rectangles
is also open in the product topology.