Just as we can define a topological vector space, we can define a "topological X" for any algebraic structure X: we demand that X also be a topological space (usually a Hausdorff space, for convenience), and that the algebraic operations on X be compatible with the topology on X by being continuous.
For a field, we'll demand that addition, negation, multiplication and inversion all be continuous on their domains. For instance, R and Q are "topological fields", since all arithmetic operations are continuous.
This is actually a useful characterization. But not all topological properties on a topological field turn out to be useful. An example which makes a nice exercise is compactness:
Let F be a topological field which is a compact space. Then F is finite.
On most structure
s (e.g. group
s), compactness extends
finiteness: many things which are true for finite objects also hold for compact objects. The above says we cannot hope for this to hold for fields: the only compact objects are
the finite ones!
Interestingly, a proof using non-standard analysis is easier and more intuitive than a "standard" proof. But (unlike the case of Sierpinski's theorem, where the non-standard proof is much easier) there is a fairly direct translation of the proof to standard analysis.