Why is differential calculus built on open sets?
For example in W. Rudin: Principles of Mathematical Analysis in every
theorem or definition regarding the derivatives of a function from
$\mathbb{R}^n \to \mathbb{R}^m$ there it always says at the beginning:
"Let $E$ be an open set in $\mathbb{R}^n$, $f$ maps $E$ to $\mathbb{R}^m$
...".
Why is the differential calculus built on the open subsets of
$\mathbb{R}^n$? What is the precise (maybe topological) explanation?
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