nLab
funcoid

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

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

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

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.