Figure 4d shows two equivalence classes that are arc-connected and two equivalence classes that are not arc connected.
In Figure 5, there are four partitions of the Hilbert curve all of whose equivalence classes are arc-connected sets.
The reader, by direct inspection, can verify that this is the only way to divide the curve of Figure 5a in two parts with the same number of points and that each part is arc-connected. Proposition 3 is derived from this.
Proposition 3: If there is a partition of a curve X with m equivalence classes, such that each equivalence class is arc-connected and has n number of points, then that partition is unique.