2
$\begingroup$

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 $X \mathrel{[ \mu]^{\ast}} Y$.

$\endgroup$
  • $\begingroup$ I've edited the question. Now it is a somehow different conjecture $\endgroup$ – porton Feb 27 '14 at 13:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.