Below is an excerpt from the paper Boundary torsion and convex caps of locally convex surfaces, in which the author defines a so-called nested partition of the circle.
I am having a hard time understanding this definition. It says that a partition $\mathcal{P}$ is nested if for every $P \in \mathcal{P}$ and $p$ in the complement $P'$ of $P$, the part $[p]$ is contained in a connected component of $P'$.
What is the geometric intuition behind this definition? Could anyone provide an example of a partition of $\mathbb{S}^{1}$ which is not nested?
For a simple non-example, take points $a,b,c,d$ which go in that order around the circle and consider a partition with $\{a,c\}$ and $\{b,d\}$ as two of its parts. This partition is not nested, since the complement of part $\{a,c\}$ has two components (the two different arcs from $a$ to $c$) and the part $\{b,d\}$ is not contained in a single one of them ($b$ and $d$ are in different components).
More generally, a nested partition is one in which different parts never "cross" like this: given distinct parts $P$ and $Q$, $Q$ must contained in just one of the arcs that makes up the complement of $P$, instead of being split up with points of $P$ in between different points of $Q$ on both sides. In fact, more precisely, a partition is non-nested iff there exist parts $P$ and $Q$ with distinct points $a,c\in P$ and $b,d\in Q$ such that these points go in the order $a,b,c,d$ around the circle. First, if there exist such points, then $Q$ is not contained in a single component of $P'$, since $b$ and $d$ will be in different components of $P'$. Conversely, suppose a partition is not nested, so there is a part $P$ and another part $Q$ which is not contained in a single component of $P'$. Pick points $b,d\in Q$ which are in distinct components of $P'$. Then $P$ must contain a point $a$ on the clockwise arc from $b$ to $d$ and also a point $c$ on the counterclockwise arc from $b$ to $d$, and then $a,b,c,d$ go in that order around the circle.