= Runge exhaustion of a slit disk
{c}
For the unit disk with the nonpositive real radius removed, compact sets $K_m=\{re^{i\theta}:1/m\leq r\leq1-1/m,\ |\theta|\leq\pi-1/m\}$, $m\geq3$, exhaust the domain. Each has connected complement: its inner and outer complementary regions join through the missing angular sector. The <polynomial Runge theorem> gives a <polynomial> within $1/m$ of a given <holomorphic function> on $K_m$. Every compact subset lies in some $K_m$, so these polynomials converge uniformly on compact subsets.
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