Given a metric space $(X,d)$ and three points $x,y,z$ in $X$, say that $y$ is *between* $x$ and $z$ if $d(x,z) = d(x,y) + d(y,z)$, and write $[x,z]$ for the set of points between $x$ and $z$.


Obviously, we have


1. $x,z\in[x,z]$;
2. $[x,z]=[z,x]$;
3. $y \in [x,z]$ implies $[x,y] \subseteq [x,z]$;
4. $w,y \in [x,z]$ implies: $w\in [x,y]$ iff $y \in [w,z]$.


My question:
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Has the family of objects with such an axiomatic “interval structure” $[\bullet,\bullet]:X \times X \to \mathcal{P}(X)$ satisfying approximately the above conditions been studied somewhere?



What happens if you strip everything but the “between” relation in metric spaces
There is a wide body of work on this in connection with the classic [De Bruijn–Erdős theorem](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(incidence_geometry)). 



> 
> **De Bruijn–Erdős Theorem.** Every set of $n$ points in the
>  plane (not all lying on the same line) determine at least $n$ lines.
> 
> 
> 


There is a beautiful conjecture of Chen and Chvátal that the De Bruijn–Erdős theorem actually holds in *every* metric space, where lines are defined using the notion of betweenness that you describe. That is, given a metric space $M$ and two points $a,b \in M$, the *line* determined by $a$ and $b$ is the set of points $c$ such that $c$ is between $a$ and $b$, or $a$ is between $c$ and $b$, or $b$ is between $a$ and $c$. Note that this definition reduces to the usual notion of lines if we use the Euclidean distance. 



> 
> **Chen-Chvátal Conjecture**. Every set of $n$ points in a metric space (not all of which lie on the same line) determine at least $n$ lines. 
> 
> 
> 


If you Google 'Chen-Chvátal Conjecture' you will find many results, including some that focus on the combinatorial aspects of betweenness. 


