Topology is in a sense generalizing the analysis on $\mathbb{R}$ to more general sets.Or in other words can we define some thing like open intervals or union of open intervals -which is called open sets- in more abstract sets.
Once we can define open sets in an abstract set -of course with the three properties !- we can generalize the concepts of continuity of functions in such sets.
In calculus or in real analysis after continuity we will define differentiability and proceed.But this definition uses either the concept of limit.
Can we define differentiability of a function between two topological spaces? or is possible only on manifolds, where functions are connected with functions between subsets of Euclidean spaces ?
