Ah, I think I see. Open sets are just how we encode information topologically. Theyre sort of the "atoms" of topology; the types of opens sets you have say how the topology behaves.

In particular open sets are neighborhoods of all of the points they contain. This means they contain all the topological information about all of their points.

Closed sets are not neighborhoods of their points in general. Eg [0,1] contains no neighborhood of 0 or 1. Then we would require knowledge of the space around those points to know how a function behaves just on [0,1].

In standard calculus this amounts to "taking the left and right limits".

Does that help?