chap-fcd-closed.tex

\chapter{Funcoids as closed sets}

Idea \cite{nlab-topogeny} by \noun{Todd Trimble}.

\fxnote{\url{https://ncatlab.org/toddtrimble/published/topogeny}
and \url{https://math.stackexchange.com/q/2681502/4876}}

\fxnote{What about the infinite products?}

\begin{thm}
The set of staroids
$\subsets X_1\times\dots\times\subsets X_n\to 2$ is
order isomorphic to co-frame of closed subsets of topological
product $\beta X_1\times\dots\times\beta X_n$.
\end{thm}

\begin{proof}
$\subsets X_1\times\dots\times\subsets X_n\to 2$ can be order-embedded to
the frame of ideals
$\mathfrak{J}(\subsets X_1\times\dots\times\subsets X_n)$ what is
dual (check!) to the frame of ideals of the distributive lattice
$\subsets X_1\otimes\dots\otimes\subsets X_n$.
This by ?? is the coproduct $\sum_i \subsets X_i$ in the category
of boolean algebras.
By Stone duality it is isomorphic to the topology of it spectrum
$\beta X_1\times\dots\times\beta X_n$.
\end{proof}

Elements of $\beta X_1\times\dots\times\beta X_n$ are closed
subsets. So every $n$-staroid one-to-one corresponds to a closed
set of $\beta X_1\times\dots\times\beta X_n$.

Note that $\beta X_1\times\dots\times\beta X_n$ is a compact
Hausdorff space (as a product of compact Hausdorff spaces).

It seems that there is an easy way to describe the above order
embedding in both directions:
\begin{gather*}
f\mapsto
\setcond{(x_1,\dots,x_n)}{
x_1,\dots,x_n\in\atoms^\mathscr{F},
x_1\times^{\mathsf{FCD}}\dots\times^{\mathsf{FCD}}x_n\sqsubseteq f};\\
X\mapsto \bigsqcup
\setcond{p_1\times^{\mathsf{FCD}}\dots\times^{\mathsf{FCD}}p_n}{
p\in X}.
\end{gather*}

\fxnote{Try to prove that composition of funcoids is isomorphic to
composition of relations $\beta A\times\beta B$.}