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Let $latex \delta$ be a proximity.
A set $latex A$ is connected regarding $latex \delta$ iff $latex \forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$.
Conjecture Set $latex A$ is connected regarding $latex \delta$ iff for every $latex a,b\in A$ there exists a totally ordered set $latex P \subseteq A$ such that $latex \min P = a$, $latex \max P = b$ and
$latex \forall a \in P \setminus \{ b \} : \left\{ x \in P \,|\, x \leqslant a \right\} \mathrel{\delta} \left\{ x \in P \,|\, x > a \right\}$.