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August 17, 2013 / porton

Pointfree funcoid induced by a locale or frame?

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.
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  1. porton / Aug 19 2013 21:45

    It seems that every locale induces a certain pointfree funcoid: First embed it into a complete boolean algebra as in http://mathoverflow.net/questions/139810/embedding-a-brouwerian-lattice-into-a-boolean-lattice and second defined closure operator on this lattice just like as I do it in with topological spaces in my book, then use this closure to define a pointfree funcoid. I am going to write on this topic more, but not now, as now I need to allocate time to check my monograph for errors and typos.

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