# uniformity

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

Author(s): Porton

\begin{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 $g = \langle f \times f \rangle^{\ast} \Delta$. \end{conjecture}

The main purpose here is to find a \emph{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 \href[this draft article]{http://www.mathematics21.org/binaries/compact.pdf}.

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

\begin{conjecture} Let $f$ be a $T_1$-separable compact reflexive symmetric funcoid and $g$ be a reloid such that \begin{enumerate} \item $( \mathsf{\tmop{FCD}}) g = f$; \item $g \circ g^{- 1} \sqsubseteq g$. \end{enumerate} Then $g = \langle f \times f \rangle^{\ast} \Delta$. \end{conjecture}

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