There is an alternate definition of the basis generated by a topology that is often useful:

Theorem 1Let be a basis for a topology on . Then if and only if

where .

*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 2A collection of subsets of is asubbasisfor 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

where

is the **projection map** defined by .

Theorem 3The 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

where

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

If then

is also open in the product topology.