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Generalized path-connectedness in proximity spaces

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
Keywords: connected
connectedness
proximity space
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) $.

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

Bibliography

*Question at math.StackExchange.com by Victor Porton


* indicates original appearance(s) of problem.
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A proposed lemma

On February 26th, 2014 porton says:

http://math.stackexchange.com/questions/691643/a-lemma-to-solve-a-conjec...

--
Victor Porton - http://www.mathematics21.org

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