# Conjecture: Connectedness in proximity spaces

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Let $\delta$ be a proximity.

A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$.

Conjecture Set $A$ is connected regarding $\delta$ iff for every $a,b\in A$ there exists a totally ordered set $P \subseteq A$ such that $\min P = a$, $\max P = b$ and

$\forall a \in P \setminus \{ b \} : \left\{ x \in P \,|\, x \leqslant a \right\} \mathrel{\delta} \left\{ x \in P \,|\, x > a \right\}$.