I can’t recall so many statements in Mathematical Analysis that hold for a certain class of possibly unbounded subsets of $\mathbb R^n$ (or $\mathbb C^n$) with the only exception being the whole space. Some examples that I have in mind are:
- The Riemann mapping theorem
- Liouville’s theorem for harmonic functions (which is somehow related to the Riemann mapping theorem). In fact, at least for $n\geq 3$, the theorem never holds for any proper open subset $\Omega\subset \mathbb R^n$, as the fundamental solution to the Laplace equation is smooth away from the origin. (edit: actually one also has to assume that $\Omega$ is not dense)
- The strict inclusion between the Sobolev spaces $W^{1,p}(\Omega)$, $W^{1,p}_0(\Omega)$ with $1\leq p<\infty$ (this one is probably more naturally seen as the contrapositive of the density of test functions in $W^{1,p}(\mathbb R^n)$, but still sounded strange to me while I was studying it)
Do you have interesting examples of theorems or statements in Analysis that hold for a certain class of subsets of $\mathbb R^n$ (open, measurable, convex,…) which does not a priori exclude $\mathbb R^n$, but for which $\mathbb R^n$ is actually the only exeptional case in which the statement breaks down?