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August 28, 2013 / porton

My study of pointfree topology

I have read The point of pointless topology today and am going to study the book Johnstone “Stone Spaces” which I purchased maybe a year or two ago.

The purpose of this study is to integrate others’ pointless topology with my theory of pointfree funcoids.

From my earlier comment on this blog:

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 define 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.

Thus (if the above considerations are correct) pointfree funcoids are a massive generalization of locales: They don’t only require the lattice of filters to be boolean but these can be even not lattices of filters at all but just arbitrary posets.

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