Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

Conjecture   Let $ f $ is a $ T_1 $-separable (the same as $ T_2 $ for symmetric transitive) compact funcoid and $ g $ is a uniform space (reflexive, symmetric, and transitive endoreloid) such that $ ( \mathsf{\tmop{FCD}}) g = f $. Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

Conjecture   Let $ f $ be a $ T_1 $-separable compact reflexive symmetric funcoid and $ g $ be a reloid such that
    \item $ ( \mathsf{\tmop{FCD}}) g = f $; \item $ g \circ g^{- 1} \sqsubseteq g $.

Then $ g = \langle f \times f \rangle^{\ast} \Delta $.

Keywords: compact space; compact topology; funcoid; reloid; uniform space; uniformity

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

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