Generalized path-connectedness in proximity spaces

Problem's Description

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 The following statements are equivalent for every endofuncoid $ \mu $ and a set $ U $:

- $ U $ is connected regarding $ \mu $.

- For every $ a, b \in U $ there exists a totally ordered set $ P \subseteq U $ such that $ \min P = a $, $ \max P = b $, and for every partion $ \{ X, Y \} $ of $ P $ into two sets $ X $, $ Y $ such that $ \forall x \in X, y \in Y : x < y $, we have $ X \mathrel{[ \mu]^{\ast}} Y $.

1. BookIs an originAlgebraic General Topology. Volume 1

   • Victor Porton

   year of publication: 2018fulltext