Is there a natural topological structure for a general field $\mathbb{K}$? If not, what are some standard structures that define a topology?
 A: Topology is often about "how close are two points". This means that topologies naturally arise when we can somehow define what does it mean for two objects to be close. Of course this is still a very relative notion, but we usually have a good sense of what it means.
When we say that a topology naturally arises from some structure then we often expect certain endomorphisms of the structure to be continuous. Addition, order preserving maps, and so on. This is because we define our basic open sets from "naturally definable subsets", such as intervals.
Addition does not induce a natural topology. This is the reason that not all the groups are topological groups, and not all the fields are topological fields.
On the other hand, order topology does induce some sort of notion of "closeness", by using cones or intervals, as basic open sets. Then given three points we can decide whether the first two are closer than the first and the third, by examining what sort of cones every pair appears in.
A: This encyclopedia article contains some basic, foundational information on topological fields.  Does it answer your question?  If not, please explain what you still want to know.
