Generalized path-connectedness in proximity spaces

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
Recomm. for undergrads: no
Posted by: porton
on: February 1st, 2014

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}

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Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

*\href [Question at math.StackExchange.com]{http://math.stackexchange.com/questions/642337/connectedness-in-proximity-spaces} by Victor Porton


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