Let $X$ be a topological space. Consider the set $B_X$ of partitions of $X$ such that every block of the partition is simply connected. How can one define a metric $d$ on $B_X$?
To help answer this, consider the following cases (of increasing complexity):
- $X = \{1,2,\ldots,n\}$, and $B_X$ contains partitions of the form $1,\ldots,t_1|t_1 +1,\ldots,t_2| \ldots|t_{-1},\ldots,n$
- $X = [0,1]$, and $B_X$ contains partitions of the interval specified by $0=t_0 <t_1 < t_2<\ldots< t_{-1}=1$
- $X = \mathbb{R}$, and $B_X$ contains partitions specified by $-\infty=t_0 <t_1 < t_2<\ldots< t_{-1}=\infty$
- $X=\mathbb{R}^n$ and the $B_X$ contains partitions formed by hyperplanes perpendicular to the axes
- $X=\mathbb{R}^n$ and the $B_X$ contains simply connected partitions
What would be an example of a non-trivial metric in these cases?
To give some context, this problem arose from trying to compare decision trees (as in machine learning) constructed from a common set of attributes: Each decision tree partitions the space of attributes with hyperplanes perpendicular to the axes.
I wonder if one can construct something similar to the Wasserstein metric or the earth mover's distance, i.e, the distance between two partitions is the minimum "cost" to convert one partition to the other. For case 1. above, if $P_1 = 1,2,3|4,5$ and $P_2 = 1,2|3,4,5$ are two partitions of $\{1,2,3,4,5\}$, then $d(P_1,P_2) = 1 $ because the $P_1$ to $P_2$ conversion involves moving just one element (3). But I'm not sure how to generalize this.