 the definition of the usual topology and
| the fact that open intervals are open relative to this topology.
|
| There is a simple characterization of bases which is frequently used
| as a definition:  A subfamily !B! of a topology !T! is a base for !T!
| iff each member of !T! is the union of members if !B!.  To prove this
| fact, suppose that !B! is a base for the topology !T! and that U in !T!.
| Let V be the union of all members of !B! which are subsets of U and suppose
| that x in U.  Then there is W i