Conjecture   Let is a -separable (the same as for symmetric transitive) compact funcoid and is a uniform space (reflexive, symmetric, and transitive endoreloid) such that . Then .

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 be a -separable compact reflexive symmetric funcoid and be a reloid such that

\item ; \item .

Then .

Conjecture   Distributivity of the lattice of funcoids (for arbitrary sets and ) is not provable in ZF (without axiom of choice).

A similar conjecture:

Conjecture   for arbitrary filters and on a powerset cannot be proved in ZF (without axiom of choice).