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January 18, 2014 / porton

Conjecture: Connectedness in proximity spaces

I’ve asked this question at math.StackExchange.com

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\}.
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