The hyperreal number system adds infinities and infinitesimals, allowing Calculus to be done using these things instead of limits (sort of like when calculus was originally invented, but with rigor).This got me thinking, could this be done with topology to create hypertopology (an equivalent but perhaps more intuitive way of doing topology)?
It should be able to translate, since hyperreals can be represented as ultrafilters of reals. Hypertopology would involve ultrafilters of the points of the topology presumably.
To define the topology, I presume you would define when two points of the hypertopology are infinitesimally close. To convert a metric space into a topology, you would simply define two hypertopology points as close whenever their distance is infinitesimal (again, ultrafilters can be applied point wise.)
Has this ever been studied? What axioms would hypertopology "closeness" need to follow to be equivalent to regular topology? Is there an axiomatic approach (not requiring the ultrafilters (the hyperreals have an axiomatic basis))?