In my book I introduce funcoids as a generalization of proximity spaces. This is the most natural way to introduce funcoids, but it was not the actual way I’ve discovered them.

The first thing discovered equivalent to funcoids was a function $latex \Delta$ (generalizing a topological space) which I defined to get a set as argument and return a filter, subject to several equalities, after the removal of superfluous equalities it has become:

  • $latex \Delta\varnothing = \varnothing$;
  • $latex \Delta(I\cup J) = \Delta I\cup\Delta J$.

(Here $latex \cup$ and $latex \cap$ may mean the join and meet on the lattice of filters ordered reverse to set-theoretic inclusion of filters, $latex \varnothing$ also denotes the improper filter.)

Then somehow (I don’t remember how exactly) I managed that this $latex \Delta$ can be reverted (like modern notion of reverse funcoid).

Somehow I have managed to define composition of funcoids.

Afterward I defined funcoids in the following cumbersome way:

Funcoids is a set of objects $latex f$ such that it is unambiguously defined by the values $latex \langle f\rangle$ and $latex \langle f^{-1}\rangle$ of functions from filters to filters such that $latex \mathcal{Y}\cap\langle f\rangle\mathcal{X}\ne\varnothing \Leftrightarrow \mathcal{X}\cap\langle f^{-1}\rangle\mathcal{Y}\ne\varnothing$.

Later I have come to the simple idea that instead of this cumbersome definition can be replaced with simply defining a funcoid as a pair of functions from filters to filters.

An other anachronism: Initially I considered funcoids as a special case of reloids conforming to the above formula. (I don’t remember whether I had an exact definition for a reloid to be a funcoid.) Later funcoids have become an independent kind of objects, rather than a special case of reloids.

Finally, I’ve replaced the funcoid on a single “universal” set with funcoids between two sets $latex A$ and $latex B$, so forming a category of funcoids.

I have not told the history of filter objects (where principal filters were equated with corresponding sets, such as $latex \varnothing$ was also the improper filter) and then removing this notion in regard of simple reverse-ordered lattice of filters with order, meets, and joins denoted differently than set-theoretic subset, intersection, and union, not to make mess between these.

Finally: If I would know the notion of proximity spaces at the time when I wrote the function $latex \Delta$ would it prevent me to discover funcoids (counting that proximity spaces is what I need and thus not continuing the research further)?

2 thoughts on “The history of discovery of funcoids

  1. The concept of “failure functions ” may be applicable. To see the possibility go to and search for “failure functions ” . You may also read how this tool has been used to prove the infinitude of primes of the form x^2 + 1.

    A.K. Devaraj ( go to You tube and search for A.K. Devaraj)

Leave a Reply

Your email address will not be published. Required fields are marked *