Decomposing the plane into intervals

A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing in every direction. That got me trying to decompose the plane into closed or open intervals. The best I could do was to make a square with two sides missing (which you can do out of either type) and form a checkerboard with the white squares missing the top and bottom and the black squares missing the left and right. That gets the whole plane except the lattice points. This seems like it must be a standard problem, but I couldn't find it on the web. So can the plane be decomposed into unit open intervals? closed intervals?

I posted this to Math Overflow and Jeff Strom gave the following answer:

Conway and Croft show it can be done for closed intervals and cannot be done for open intervals in the paper:

Covering a sphere with congruent great-circle arcs. Proc. Cambridge Philos. Soc. 60 1964 787–800.

Seems pretty easy. We're going to cover the complex plane instead since that's obviously equivalent to covering $\mathbb{R}^2$. Start with the family of unit length line segments $I_\theta = \{a\cdot e^{i\theta} \mid a \in (0,1] \}$, $\theta \in (0,2\pi)$. Then define $I_{\theta,n}$, $n \in \mathbb{N}_0$ to be $I_\theta$ translated a length of $n$ in the direction of $\theta$. This collection covers the entire complex plane except the ray $[0,\infty)$ which we cover with the half-open intervals $[k,k+1)$.

Armed with this idea, it should be easy to see how to for any $k$ fill $\mathbb{R}^k$ with intervals of unit length that "point in all directions".

FYI: \mathbb{N}_0 breaks stuff. You have to escape the underscore: \mathbb{N}\_0.