Have you studied math? No mathematician would call algebraic topology discrete math, nor for that matter almost anything else on that list.
I'd have to ask around but I know a few algebraic topologists and I'd bet that if pressed to describe their work as more-continuous or more-discrete they'd first tell you that this is a silly way to classify subfields in math and then they'd says it's probably more on the discrete side since it's all about categorizing topological spaces based on whether they have certain discrete properties:
- separable - countable - metric - compact
Sure the spaces themselves might be continuous, but topology is for telling those spaces apart based on where and how continuity fails. It's a discretization of things formerly suspected to be the the same.
There's something about real analysis that makes me uncomfortable. I've been struggling for years to put my finger on it. Whatever it is, topology doesn't have it.
I had mistakenly decided that it was an overappreciation of continuity, but I think it must be something else.
Something about the homework in Real Analysis left me feeling angry. Not because it was difficult or presented poorly, but because it was somehow... untrustworthy? As if my betters had decided which ideas were the good ones and the only thing left for me to do was optimize along the one dimension that they had assigned me. I realize that this is nonsense, but I can't seem to shake it.