# soft question – How to balance generality and intuition while self-studying?

Currently I don’t have access to high-level math classes (say from a college) so I resort to self-study. In my experience so far one challenge with learning new topics is balancing intuition and a desire for generality.

Consider how much of the basic topology required for real analysis can be developed either in arbitrary (perhaps Hausdorff) topological spaces or in metric spaces. As a sucker for generality, I learned about topological and metric spaces before calculus. But I think going straight for topology before analysis helped contextualize much of the topology present in calculus, so when I got to reading baby Rudin for example the first few chapters were pretty easy.

Part of the reason I felt comfortable learning topology before analysis was I had some idea what it was supposed to be generalizing (“nearness”, I guess), so I could picture some of what was going on in, say, $$mathbb{R}^2$$ to guide my intuition and proofs. But a lack of familiarity with topological arguments in any context meant it wasn’t so easy.

At the same time, I don’t find explicit examples helping much when generalizing them, because often the details and proofs are made specific to the context given and I don’t recognize them as special instances of a general theory. Sometimes the perspective is actually completely different – for example, my first taste of multivariable real integration has been with differential forms. But while researching vector fields, I can’t help but notice the approach using differential forms doesn’t seem conceptually similar to integration over vector fields. The connection between $$1$$-forms and vector fields for example seems to be that the linear functional assigned by the $$1$$-form is dual to the vector from the field (is this correct?). I don’t think I would have made this observation about $$1$$-forms “naturally” after having learned about line integration.

In other words: I’m not confident a rigorous calculus course would have made learning topology any easier.

So how do I effectively balance intuition with generality and formality? How might I go about learning very abstract topics if I don’t know what they’re used for? Currently I just barge through the abstract (not that I don’t enjoy it) then attempt to reconcile the two later. Or is my attempt at shooting straight for “topological spaces” before “analysis” not a good method of studying?