Something that I've found interesting is that the domain of a derivative always seems to be phrased in terms of open sets, even when the original function is defined on a closed set. It's like taking the derivative removes exactly the bounds from the domain. I wonder if one-hole contexts are to blame for that.
No, because in discrete differentiation like this, there is no limiting process where openness matters, and open/closedness is completely trivial in discrete spaces.
The "one-hole" is in the discrete types, not the potentially continuous values.
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".
> 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.
This is an interesting perspective, thank you. It reminds me of NAND gates
