Start with the interval $[0,1)$ embedded in the $x$-axis. Fill in the closed unit disc minus the point $(1,0)$ using half-open intervals that point inward. Add the interval $[1,2)$ in the $x$-axis. Fill in the closed disc of radius 2 minus the point $(2,0)$ using half-open intervals that point inward. Repeat: For each $n \geq 0$, you get a closed disc of radius $n$ minus a point. The union of these is the plane.

Here is a solution to the half-open interval problem:

1. Start with the interval from $(0,0)$ to $(1,0)$ that contains the endpoint $(0,0)$.
2. Fill in the closed unit disc minus the point $(1,0)$ using half-open intervals that point inward.
3. Add the interval from $(1,0)$ to $(1,1)$ that contains the endpoint $(1,0)$.
4. Fill in the closed disc of radius $\sqrt{2}$ minus the point $(1,1)$ using half-open intervals that point along inward tangents. Now we have segments in all directions.
5. Fill in the ray $\{ (a,a) \mid a \geq \sqrt{2} \}$ using outward pointing radial segments.
6. Fill in the rest of the plane using inward pointing radial segments arranged in concentric annuli (with slits at $\arg z = \frac{\pi}{4}$).