The Topological Field of Real Numbers We’ve defined the
topological space we call the real number line as the completion
of the rational numbers as a uniform space. But we want to be
able to do things like arithmetic on it. That is, we want to put
the structure of a field on this set. And because we’ve also got
the structure of a topological space, we want the field
operations to be continuous maps. Then we’ll have a topological
field, or a “field object” (analogous to a group object) in the
category of topological spaces. Not only do we want the field
operations to be continuous, we want them to agree with those on
the rational numbers. And since is dense in (and similarly is
dense in ), we will get unique continuous maps to extend our
field operations. In fact the uniqueness is the easy part, due
to the following general property of dense subsets. Consider a
topological space with a dense subset . Then every point has a
sequence with . Now if and are two continuous functions which
agree for every point in ,