For context: I’ve taken point-set topology, am now taking algebraic topology and differential topology.
I did well in point-set. But it was honestly very hard. I’m used to working with abstract concepts, but we started with very bare-bones definitions (e.g. a topology is a set of subsets such that…). Nothing made sense until we started working with metric spaces. Then it all felt natural.
Except metric spaces are obviously very nice. Many properties don’t generalize beyond metric spaces. But I have no working intuition of them, my reasoning feels constrained to metric spaces. I can obviously prove things for general topological spaces, but sometimes I can’t imagine it.
The worst part is the following: sometimes I completely miss the motivation for a definition. For example, compactness. I know why in the Euclidean metric space a set is compact <-> it is closed and bounded. Sure. But now my intuition operates on closed and bounded sets, even though it doesn’t generalize past the standard topology in R\^n. I would’ve never thought to define a compact set with finite subcovers.
Or the separation axioms. They feel arbitrary (although admittedly we’ve only really worked with Hausdorff spaces thus far).
I very much need your advice. I enjoy topology very much despite my struggles so I’m willing to put in the work to make it feel natural.