I have shown in my research monograph that topological (even pre-topological) spaces are essentially (via an isomorphism) a special case of endo-funcoids.
It was natural to suppose that locales or frames induce pointfree funcoids, in a similar way.
But I just spent a few minutes on defining the pointfree funcoid corresponding to a locale or frame and found that it is at least not quite trivial (that is I failed to define it).
If you have any ideas about how pointfree funcoids correspond to locales or frames, please comment. I don’t know this.
As for now the situation in the ongoing research in general topology is the following:
- Topological space is a special case of endo-funcoids. The topologists should move their attention from topological spaces to funcoids in the same way as analysis has moved from real to complex numbers.
- In pointfree topology (the theory of locales and frames) this transition however has not (yet?) happened. We may study pointfree funcoids and locales/frames in parallel. Both are expected to be useful.