Let $(X,d)$ be a metric space and let $x,y \in X$ be points. We will define the metric segment between the points $x,y$ as the following set:
$\left [ x,y \right ]=\left \{ z \in X : d(x,z)+d(z,y)=d(x,y) \right \}$
Also, we will say that a set $C \subseteq X$ is convex if for all $x,y \in C$ it holds true that $\left [ x,y \right ] \subseteq C$.
It can be easily seen that $\emptyset$ and $X$ are convex and that the intersection of any two convex sets is a convex set.
We will call a family $\mathcal{F} \subseteq \mathcal{P} (X)$ 'special' if the following is satisfied:
- $\emptyset, X \in \mathcal{F}$
- $\forall U, V \in \mathcal{F} : U \cap V \in \mathcal{F}$
Furthermore, we will call such a family $\mathcal{F}$ metrizable if there exists a metric $d: X \times X \rightarrow \mathbb{R}$ such that the set of all convex sets in $(X,d)$ is equal to $\mathcal{F}$.
Denote by $a(n)$ the overall number of metrizable special families that can be defined on a set of $n$ elements.
My question is how can we write a code to enumerate the sequence $a(n)$?
How would we even go about checking whether a function $d: X \times X \rightarrow \mathbb{R}$ on a finite set $X$ is a metric?
Note: The number of special families on $n$ is given in OEIS as A102894. As an example, the special family $\left \{\emptyset, \left \{ 1 \right \}, \left \{2 \right \}, \left \{3 \right \}, \left \{ 1,2 \right \}, \left \{2,3 \right \}, \left \{1,2,3 \right \} \right \}$ is metrizable because it precisely agrees with the set of all convex sets in the metric space $(\left \{1,2,3 \right \}, d)$ where $d$ is the standard Euclidean metric on $\mathbb{R}$. Also, as I understand, there is this rather simple code to evaluate the number of (non-necessarily metrizable) special families on a set of $n$ elements:
b[n_] :=
Length[Select[Subsets[Subsets[Range[n]]],
And[MemberQ[#, {}], MemberQ[#, Range[n]],
SubsetQ[#, Intersection @@@ Tuples[#, 2]]] &]]
We would just need to incorporate the metrizability condition, but I am clueless on how to go about this.
