Consider the following statement (contrived to highlight an unknown piece of terminology):
For any $n$-dimensional space $\mathbb{R}^n$, a finite number of $(n-1)$-dimensional hyperplanes in a general arrangement will separate $\mathbb{R}^n$ into a finite number of $n$-dimensional regions. Let $d_i \in \mathbb{Z}$ be region $i$'s "regional distance" from a reference line (that is, of course, $1$ dimensional). The reference line can be assumed not to be a subset of any of the hyperplanes.
By "regional distance" I mean the minimum number of $n$-dimensional regions that need to be "traversed" to get from any point on the line to a point in the given region, where $d_i = 0$ if region $i$ intersects the reference line. Obviously I have not defined "regional distance" well, but I expect it has a term (or there is a term for a very similar concept) that I'm unaware of. What I hope is that someone reading this knows the right term for me to google.
To illustrate
which shows $d_i$ for all regions, for an example $n=2$ case. As shown, any region that is adjacent to a region that intersects the reference line has "regional distance" 1, and any region that is adjacent to a region that is adjacent to a region that intersects the reference line has "regional distance" 2.
EDIT: Fixed error in image.