Common generalizations of convergences and funcoids

I am reading the book “Convergence Foundations of Topology” by Szymon Dolecki and Frédéric Mynard. (Well, I am more skimming than reading, I may read it more carefully in the future.)

After reading about a half of the book, I tried to integrate my theory of funcoids with their theory of convergences.

And I noticed, that if I define convergences induced by funcoids following “Convergence of funcoids” chapter in my book, then convergences induced by (reflexive) funcoids are pretopologies (a narrow subclass of convergences). So convergences appear to be not a special case of funcoids. Certainly it seems that in the other direction funcoids are not a special case of convergences.

So we have a (probably difficult) problem: Find a common generalization of funcoids and convergences!

3 comments

  1. I propose the following ideas for this common generalization:

    1. To every set we associate an isotone (and in some sense preserving finite joins) collection of filters.
    2. To every filter we associate an isotone (and in some sense preserving finite joins) collection of filters.
    3. Consider pointfree funcoids between isotone families of filters.
  2. Convergence could be approached by having two simultaneous twin funcoids – one starting at “0” and going “up” and the other starting at “1” and going “down”.

    Just a thought.

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