For all $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such that $a'\in A$ and $c'\in C$. What can we conclude about $A,C$?

The $\mathbb R^n$ space is partitioned into three sets $A,B,C$.

Given:

1. $B$ is convex.
2. $A,C$ are non empty.
3. For all open segment $(a,c)$ where $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such that $a'\in A$ and $c'\in C$.

What can we conclude about set $B$?

Source: I love to think about some uncommon math scenarios. This question is self-made.

Motivating example: Consider $\mathbb R$. $B$ must be connected as it is convex. Condition 3 might imply that $A$ is disjoint from $C$, which means that $A\cup C$ is not a connected set. So I think $B$ must be a single point or a closed interval.

Hypothesis: $A,C$ must be open half-spaces.