Talk:Stone duality

In the section The lattice of open sets I read that the lattice of open sets of a topological space is complete. I think this is not true in general, for the intersection of an arbitrary family of open sets need not be open. --fudo 12:17, 12 December 2006 (UTC)[reply]

OK, I think I got it. I'll write here my reasoning instead of deleting my previous comment so that it may help anybody else not to make the same mistake: Even if the intersection of an arbitrary family of open sets of a topological space need not be open, such a family DOES have an infimum in the open set lattice, which is the interior of its intersection. --fudo 12:48, 12 December 2006 (UTC)[reply]

While it's indeed true that the open sets of a topological space ordered by inclusion form a complete lattice, how does this bear on Stone duality? It seems like an irrelevant distraction of interest only in certain application areas of Stone duality. --Vaughan Pratt (talk) 19:46, 26 January 2008 (UTC)[reply]

Proposal to reorganize this article

This article could be made much clearer by postponing its dependence on the machinery of Isbell's locale theory and Scott's domain theory or moving those applications to a separate article. It suggests incorrectly that these are prerequisites for Stone duality, and it places an unnecessary burden on readers by forcing them to consult other articles to absorb perspectives that take the reader far beyond Stone duality itself.

The essential prerequisites for Stone duality are Boolean algebras up to any definition of ultrafilter (e.g. as a homomorphism to the two-element Boolean algebra 2), and point-set topology up to the notions of basis, T₂ or Hausdorff space, compactness, and connectedness. The Stone space of a Boolean algebra is the topological space generated by its ultrafilters as a basis for the topology, i.e. the result of closing the set of ultrafilters under arbitrary union (they are already closed under finite union and finite intersection because the pointwise Boolean combination of homomorphisms is itself a Boolean homomorphism.) The spaces arising in this way together with all continuous functions between them then form a category which is dual to, meaning equivalent to the opposite of, the category of Boolean algebras. Moreover every such space turns out to be compact, Hausdorff, and totally disconnected. Remarkably, and arguably the nicest observation in Stone's 1936 paper introducing this duality (albeit without noting the induced contravariance in the morphisms), up to isomorphism every totally disconnected compact Hausdorff space, or Stone space as now called, arises from some Boolean algebra in this way, establishing the duality of Boolean algebras and Stone spaces. None of this requires the notions of either locale or Scott domain.

Focusing on the "founding" example of Stone duality in this way serves two purposes: it minimizes the prerequisites for an initial grasp of the concept, and it is faithful to the historical development. One can then move on to other instances of Stone duality such as that between distributive lattices and Priestley spaces (using Priestley's improved characterization of the class of spaces originally identified more obscurely by Stone in 1937), symmetric examples such as complete semilattices which admit nontrivial topology on both sides of the duality, Pontrjagin duality (which already has its own article), and however many other examples from Johnstone's Stone Spaces and other sources make sense for a Wikipedia article and are not covered elsewhere in Wikipedia. The point can also be made that topology is unavoidable in dualities between algebras, in the sense that nondiscrete topology can always be identified on one or both sides of the duality, e.g. the dual of sets as discrete spaces is complete atomic Boolean algebras as algebras admitting Stone topology, more generally posets as dual to profinite distributive lattices, etc.

If those responsible for this article agree with this criticism I'd be happy to work with them to improve it suitably. In particular I would propose moving the bits having locales and domain theory as prerequisites at least to after the introductory explanation of the part Stone was responsible for, as applications of Stone duality. However it is no small undertaking to learn about locales and domain theory, each a substantial subject in its own right, and it might make more sense to move these applications to separate articles addressing audiences having those interests (which can easily link to the main Stone duality article). What doesn't make sense is having the main Stone duality article thoroughly entangled with that material from the outset, which neither motivated the original concept (locales and Scott domains came several decades after Stone duality) nor is needed to grasp the basic idea. --Vaughan Pratt (talk) 19:35, 26 January 2008 (UTC)[reply]

too much blabbing

This article is written like a text book for people who don't have the required prerequisites, iow, it's pointless and useless. This is not the place to apologize to a non expert that math is hard.--345Kai (talk) 21:17, 2 September 2014 (UTC)[reply]