ecause not every
| family of sets is the base for a topology.  For example, let X
| consist of the integers 0, 1, 2, let A consist of 0 and 1, and
| let B consist of 1 and 2.  If !S! is the family whose members
| are X, A, B, and the void set, then !S! cannot be the base
| for a topology because:  by direct computation, the union
| of members of !S! is always a member, so that if !S! were
| the base of a topology that topology would have to be !S!
| itself, but !S! is not a topology because A |^| B is not
|