# nLab funcoid

Let $𝔉\left(A\right)$ and $𝔉\left(B\right)$ be the reverse lattices of filters for the sets $A$ and $B$.

Then a funcoid from $A$ to $B$ consists of functions $\alpha :𝔉\left(A\right)\to 𝔉\left(B\right)$ and $\beta :𝔉\left(B\right)\to 𝔉\left(A\right)$ such that

$\forall 𝒳\in 𝔉\left(A\right),𝒴\in 𝔉\left(B\right):\left(𝒴\sqcap \alpha 𝒳\ne {0}^{𝔉\left(A\right)}⇔𝒳\sqcap \beta 𝒴\ne {0}^{𝔉\left(B\right)}\right).$\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 $𝒳$ on $A$ and every filter $𝒴$ on $B$, the filter (on $B$) generated by the union of $𝒴$ and $\alpha \left(𝒳\right)$ is not the entire power set $𝒫\left(B\right)$ iff the filter (on $A$) generated by the union of $𝒳$ and $\beta \left(𝒴\right)$ is not the entire power set $𝒫\left(A\right)$.

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 November 24, 2012 03:24:27 by Victor Porton (77.126.171.144)