Hi I'm working on Rudin's real analysis textbook and I motivate the proofs about topology using $R$ $R^2$ and $R^3$ in my mind but I know he proves it for $R^n$. Even though I know if it works for the latter it should work the former, I am wondering if I should totally forget about the geometric intuition and solely work with logic and definitions for proving things more accurately in my mind as well. Or does he also motivate them with the basic metric spaces and just generalizes it to $R^n$?
2026-05-06 06:20:43.1778048443
Motivating topological proofs
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You should definitely not forget about geometric intuition. It is a very powerful tool, even in high dimensions. $\mathbb R^n$ is much like $\mathbb R^2$ or $\mathbb R^3$, it just has a few extra dimensions thrown in.