Editing General topology (section)

====Topologies on the real and complex numbers====
There are many ways to define a topology on '''R''', the set of [[real number]]s.  The standard topology on '''R''' is generated by the [[Interval (mathematics)#Definitions|open intervals]].  The set of all open intervals forms a [[base (topology)|base]] or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the [[Euclidean space]]s '''R'''<sup>''n''</sup> can be given a topology.  In the usual topology on '''R'''<sup>''n''</sup> the basic open sets are the open [[Ball (mathematics)|ball]]s.  Similarly, '''C''', the set of [[complex number]]s, and '''C'''<sup>''n''</sup> have a standard topology in which the basic open sets are open balls.

The real line can also be given the [[lower limit topology]].  Here, the basic open sets are the half open intervals <nowiki>[</nowiki>''a'', ''b'').  This topology on '''R''' is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology.  This example shows that a set may have many distinct topologies defined on it.