Direct proof of a theorem about compact funcoids

Importance: Medium ✭✭
Author(s): Porton, Victor
Subject: Topology
Recomm. for undergrads: no
Posted by: porton
on: February 8th, 2014

\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}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

Bibliography

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

Victor Porton. \href[Compact funcoids]{http://www.mathematics21.org/binaries/compact.pdf}


* indicates original appearance(s) of problem.