There are already rich geometric interpretations which provide useful intuition for that generalizes, rather than just demonstrating mechanical details.

In any event reading the work of philosophers has certainly made me a better thinker and mathematician. Without a broad base of thought one can easily get sucked into many intellectual traps and tar pits. In my case, I found Penelope Maddy and Wittgenstein to be helpful when I was figuring out what math "is".

Its an informative exercise to see how many of the standard rules of calculus can be derived using just this rule.

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?

Differential fields are (as I understand it) a hot topic at the moment. Maybe there is some interesting crossovers.

If portability is not a concern, you can pick up used high end Nikon DSLRs and F mount lenses very cheaply right now. Nikon is going all in on mirrorless now so this stuff is "last gen" hence cheap.

Otherwise, just avoid Canon. Theyre becoming increaingly scummy and you probably dont wanna get caught locked into their system.

If apple starts letting us swap the glass out one day we might be have a fight but currently I just dont see it being one at all.

Eg Stuff about types and endofunctors feels like trivial examples using the jargon just to use it. OTOH discovering the Top->Grp functor in algebraic topology is kind of unexpected and revealing.