Skip to content
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.

One Comment

Leave a Comment
  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 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.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: