I was pondering about the definition of topology and how it lets us define continuous function. Before I go into my question, I would like to share how I understand topology over a set. To speak about regularity/ continuity of a function, it would not be enough to talk about what happens pointwise in the domain and codomain but one would need to speak of what happens near points (the subsets we choose define which near points we consider).
So, I thought, suppose we have an indexed family of functions $\{f_{\alpha} \}$ where $f_{\alpha}: X \to Y$, then would it be a useful pursuit to consider the continuity of the collection of functions as a whole? So, the topology we use on the domain and codomain would be some subset of $P(P(X))$ and $P(P(Y))$.
What are some interesting theorems which could be found as a result of defining continuity on a family of functions like this?
When I tried pondering about it, I think so maybe this may help in some theorems related to uniform convergence but I am not sure.
Sidenote: I think that this could be even done for collection of a collection of function by considering the powerset of the power set of the power set of the sets in domain and codomain.
