Runge exhaustion of a slit disk

For the unit disk with the nonpositive real radius removed, compact sets $K_m=\{r{e}^{i\theta }:1/m\le r\le 1-1/m,|\theta |\le \pi -1/m\}$, $m\ge 3$, 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.