We can express $\mathbb{R}^3$ as a disjoint union of circles. There are some constructive ways of doing this, although it's easier to construct them sequentially by transfinite induction, applying the following step to each point $P_{\alpha}$ in the well-ordering of $\mathbb{R}^3$ by the initial ordinal of $2^{\aleph_0}$:

- Suppose the point $P_{\alpha}$ has not already been covered (otherwise, move on to the next point);
- Choose a plane passing through that point, such that the direction of the normal vector is distinct from all previous planes;
- This plane intersects the union of all existing circles in fewer than $2^{\aleph_0}$ points, so we can draw a circle on that plane passing through $P_{\alpha}$ and disjoint from all existing circles.

We can also express $\mathbb{R}^3 - \ell$ as a disjoint union of pairwise linked circles, where $\ell$ is an arbitrary line. Specifically, we take the stereographic projection of the Hopf fibration, and remove the `circle' passing through the point at infinity.

This suggests that there may be a mutual generalisation of both results, namely that $\mathbb{R}^3$ can be expressed as a disjoint union of pairwise linked circles. Can anyone prove or disprove this conjecture?