# uniform space

## Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

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 .