Is it possible to characterize function properties such as Riemann integrability and being of bounded variation on an interval [a,b]⊆R or even R by defining specific topologies $T_1$ and $T_2$ on the domain and codomain, respectively, such that a function f:(R,$T_1$)→(R,$T_2$) possesses the property if and only if it is continuous with respect to these topologies?
We know that there exist topologies where continuity is equivalent to monotonicity, even fucntions and preorders. Are there analogous results for:
*Riemann Integrability?
*Functions of Bounded Variation?
Given the known result by Robert Geroch ("No Topologies Characterize Differentiability as Continuity"), which addresses differentiability, I am interested in whether similar characterizations or obstructions exist for integrability and bounded variation. A post dedicated for analogous question on Differentiability is here: ( Topologically, is there a definition of differentiability that is dependent on the underlying topology, similar to continuity? ).
What are some known theorems, counterexamples, or general strategies for approaching the problem of characterizing function properties using topological continuity? Are there specific types of topologies that have been explored in this context?
