Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function

complex-analysis real-analysis general-topology derivatives

1) Let $\{B_{r_j}(a_j)\}$ be a countable collection of open balls (radius $r_j$, centre $a_j$) whose union is the complement of $E$. Take $f(x) = \sum_j c_j g(\|x - a_j\|/r_j)$ for some smooth function $g$ which is nonzero on $[0,1)$ and $0$ on $[1,\infty)$ and a suitable sequence of positive numbers $c_j$.

2) The zeros of a nonconstant analytic function form a discrete set.

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