Topology

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) $.

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 $.

Keywords: connected; connectedness; proximity space

Posted by porton
updated February 26th, 2014
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Geometry

Dirac's Conjecture ★★

Author(s): Dirac

Conjecture   For every set $ P $ of $ n $ points in the plane, not all collinear, there is a point in $ P $ contained in at least $ \frac{n}{2}-c $ lines determined by $ P $, for some constant $ c $.

Keywords: point set

Posted by David Wood
updated January 22nd, 2014
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