Open Problem Garden

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

$( \mathsf{\tmop{FCD}}) g = f$

$g \circ g^{- 1} \sqsubseteq g$

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

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

Posted by porton
updated February 8th, 2014

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A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$ .