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

Then a funcoid from AA to BB consists of functions α:𝔉(A)𝔉(B)\alpha\colon \mathfrak{F}(A) \to \mathfrak{F}(B) and β:𝔉(B)𝔉(A)\beta\colon \mathfrak{F}(B) \to \mathfrak{F}(A) such that

𝒳𝔉(A),𝒴𝔉(B):(𝒴α𝒳0 𝔉(A)𝒳β𝒴0 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\mathcal{X} on AA and every filter 𝒴\mathcal{Y} on BB, the filter (on BB) generated by the union of 𝒴\mathcal{Y} and α(𝒳)\alpha(\mathcal{X}) is not the entire power set 𝒫(B)\mathcal{P}(B) iff the filter (on AA) generated by the union of 𝒳\mathcal{X} and β(𝒴)\beta(\mathcal{Y}) is not the entire power set 𝒫(A)\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 nnLab, Toby Bartels thinks that there might be something to it.

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