A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$.

Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$. However, can all these paths be constructed so that they don't cross?.

**Note**: If $S$ is simply connected, then by the Riemann mapping theorem, we can map $S$ to the unit disk, and connect $x$ to points in the boundary by (shifted) wheel spokes.
In the non-simply connected case, the paths have to somehow "avoid" the holes..
Perhaps there's some differential equation with whose solution produces such paths?
(maybe something like a laplace equation solution with boundary value f=1 at x and at the holes, and f=0 at the boundary, and paths following the gradient of f..?)