What are some nontrivial topological spaces with rich structures (and which possibly occur in higher mathematics) to keep in mind as an example to test various topological theorems and/or to build intuition behind them?
I only have the usual metric space topology on $\mathbb{R}^i$ where $1 \leq i \leq 3$ in my mind (besides the toy examples which are given in the book for that particular theorem and which are mostly too shallow in structure to be used for building intuition for other theorems) while checking various topological theorems or guessing correctness of some topological facts ; but that don't exactly provide me with the correct intuition for the generality/limitedness of the theorem.
What are some of the topological spaces which I should internalize and study properly to have a good intuition re: point set topology (as in, normal point set topology; not into too much arcane rarely explored avenues of it)