Generalized path-connectedness in proximity spaces ★★

Author(s): Porton

Let $\delta$ be a proximity.

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

\begin{conjecture} The following statements are equivalent for every endofuncoid $\mu$ and a set $U$: \begin{enumerate} \item $U$ is connected regarding $\mu$. \item 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 $X \mathrel{[ \mu]^{\ast}} Y$. \end{enumerate} \end{conjecture}

Keywords: connected; connectedness; proximity space

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