funcoid

Let $\mathfrak{F}(A)$ and $\mathfrak{F}(B)$ be the reverse lattices of filters for the sets $A$ and $B$.

Then a **funcoid** from $A$ to $B$ consists of functions $\alpha\colon \mathfrak{F}(A) \to \mathfrak{F}(B)$ and $\beta\colon \mathfrak{F}(B) \to \mathfrak{F}(A)$ such that

$\forall\mathcal{X}\in\mathfrak{F}(A),\mathcal{Y}\in\mathfrak{F}(B): (\mathcal{Y}\sqcap\alpha \mathcal{X} \ne 0^{\mathfrak{F}(A)} \Leftrightarrow \mathcal{X}\sqcap\beta \mathcal{Y} \ne 0^{\mathfrak{F}(B)}).$

That is, for every filter $\mathcal{X}$ on $A$ and every filter $\mathcal{Y}$ on $B$, the filter (on $B$) generated by the union of $\mathcal{Y}$ and $\alpha(\mathcal{X})$ is not the entire power set $\mathcal{P}(B)$ iff the filter (on $A$) generated by the union of $\mathcal{X}$ and $\beta(\mathcal{Y})$ is not the entire power set $\mathcal{P}(A)$.

Funcoids are studied by Victor Porton, an independent researcher, as an approach to general topology, as a massive generalisation of proximity spaces, pretopological spaces, etc; no other mathematicians are known to work on them. Of the regular contributors to the $n$Lab, Toby Bartels thinks that there might be something to it.

Revised on July 17, 2013 17:50:15
by Anonymous Coward
(84.228.168.151)