From Wikipedia (Note that below I corrected a place which I think is a typo):
A set system $(E, F)$ is a collection $F$ of subsets of a ground set $E$. $(E, F)$ is said to have the Interval Property, if $A, B, C ∈ F$ with $A ⊆ B ⊆ C$, then, for all $x ∈ E \setminus C$, $A\cup \{x\} ∈ F$ and $C\cup \{x\} ∈ F$ implies $B\cup\{x\} ∈ F$.
Does the name "the Interval Property" suggest it comes from some properties of intervals on $\mathbb{R}$? Otherwise, how well can the interval property be understood?
Thanks and regards!
This is my guess. The interval property generalizes a certain behavior of intervals in $\mathbb{R}$. If $A$ is something like an interval and $A \cup \{ x \} $ is also something like an interval then either $x \in A$ or $x$ is something like an endpoint for $A$. For example $(0, 1) \cup \{ 1 \} = (0, 1]$, another interval.