This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of choice.

• $\mathbb{R}^3$ is the union of disjoint circles, all with the same radius. Proof: Well-order the points in type $\frak{c}$. At stage $\alpha$, consider the $\alpha^{\rm th}$ point. If it is not yet covered, we may find a unit circle through this point that avoids the fewer-than-$\frak{c}$-many previously chosen circles. QED

• $\mathbb{R}^3$ is the union of disjoint circles, with all different radii. Proof: at stage $\alpha$, find a circle of a previously unused radius that avoids the fewer-than-$\frak{c}$-many previously chosen circles. QED

• $\mathbb{R}^3$ is the union of disjoint skew lines, each with a different direction. Proof: at stage $\alpha$, put a line through the $\alpha^{\rm th}$ point with a different direction than any line used previously. QED (Edit: it appears that one might achieve this one constructively by using the $z$-axis and a nested collection of hyperboloids, which unless I am mistaken would give lines of different directions.)

• $\mathbb{R}^3$ is the union of disjoint lines, each of which pierces the unit ball.

• And so on.

The excellent article Jonsson, M.; Wästlund, J., Partitions of (\mathbb{R}^3) into curves, Math. Scand. 83, No. 2, 192-204 (1998). ZBL0951.52018. JStor, proves among other things that $\mathbb{R}^3$ can be partitioned into unlinked unit circles, either all with the same radius or all with different radii, and they have a very general theorem about partitioning $\mathbb{R}^3$ into isometric copies of any family of continuum many real algebraic curves.

Meanwhile, there are also a few concrete constructions, which do not use the axiom of choice. For example, Szulkin (see theorem 1.1 in the Jonsson, Wästlund article) shows that $\mathbb{R}^3$ is the union of disjoint circles by a completely explicit method. And there are others. But my question is not about these cases where there is a concrete construction. Rather, my question is:

This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of choice.

• $\mathbb{R}^3$ is the union of disjoint circles, all with the same radius. Proof: Well-order the points in type $\frak{c}$. At stage $\alpha$, consider the $\alpha^\text{th}$ point. If it is not yet covered, we may find a unit circle through this point that avoids the fewer-than-$\mathfrak{c}$-many previously chosen circles. QED

• $\mathbb{R}^3$ is the union of disjoint circles, with all different radii. Proof: at stage $\alpha$, find a circle of a previously unused radius that avoids the fewer-than-$\mathfrak{c}$-many previously chosen circles. QED

• $\mathbb{R}^3$ is the union of disjoint skew lines, each with a different direction. Proof: at stage $\alpha$, put a line through the $\alpha^\text{th}$ point with a different direction than any line used previously. QED (Edit: it appears that one might achieve this one constructively by using the $z$-axis and a nested collection of hyperboloids, which unless I am mistaken would give lines of different directions.)

• $\mathbb{R}^3$ is the union of disjoint lines, each of which pierces the unit ball.

• And so on.

The excellent article Jonsson, M.; Wästlund, J., Partitions of $\mathbb{R}^3$ into curves, Math. Scand. 83, No. 2, 192–204 (1998). ZBL0951.52018. JStor, proves among other things that $\mathbb{R}^3$ can be partitioned into unlinked unit circles, either all with the same radius or all with different radii, and they have a very general theorem about partitioning $\mathbb{R}^3$ into isometric copies of any family of continuum many real algebraic curves.

Meanwhile, there are also a few concrete constructions, which do not use the axiom of choice. For example, Szulkin (see theorem 1.1 in the Jonsson, Wästlund article) shows that $\mathbb{R}^3$ is the union of disjoint circles by a completely explicit method. And there are others. But my question is not about these cases where there is a concrete construction. Rather, my question is:

The excellent article M. Jonsson, J. Wästlund, "Partitions of $\mathbb{R}^3$ into curves" Math. Scand. 83 (1998) 192-204 proves among other things that $\mathbb{R}^3$ can be partitioned into unlinked unit circles, either all with the same radius or all with different radii, and they have a very general theorem about partitioning $\mathbb{R}^3$ into isometric copies of any family of continuum many real algebraic curves.

The excellent article Jonsson, M.; Wästlund, J., Partitions of (\mathbb{R}^3) into curves, Math. Scand. 83, No. 2, 192-204 (1998). ZBL0951.52018. JStor, proves among other things that $\mathbb{R}^3$ can be partitioned into unlinked unit circles, either all with the same radius or all with different radii, and they have a very general theorem about partitioning $\mathbb{R}^3$ into isometric copies of any family of continuum many real algebraic curves.

1. Joseph O'Rourke: Filling $\mathbb{R}^3$ with skew lines

2. Zarathustra: Is it possible to partition $\mathbb{R}^3$ into unit circles?

3. Benoît Kloeckner: Is it still impossible to partition the plane into Jordan curves without choice?

1. Joseph O'Rourke: Filling $\mathbb{R}^3$ with skew lines

2. Zarathustra: Is it possible to partition $\mathbb{R}^3$ into unit circles?

3. Benoît Kloeckner: Is it still impossible to partition the plane into Jordan curves without choice?