Page "Base (topology)" Paragraph 4

If a collection B of subsets of X fails to satisfy either of these, then it is not a base for any topology on X.

( It is a subbase, however, as is any collection of subsets of X.

) Conversely, if B satisfies both of the conditions 1 and 2, then there is a unique topology on X for which B is a base ; it is called the topology generated by B.

( This topology is the intersection of all topologies on X containing B.

) This is a very common way of defining topologies.

A sufficient but not necessary condition for B to generate a topology on X is that B is closed under intersections ; then we can always take B < sub > 3 </ sub > = I above.