Dear Vaughan, The problem with these representations (Nachbin, Priestley, ...) is that they are not completely functorial. The natural morphisms (order-preserving continuous maps) between the Nachbin or Priestley spaces correspond not to arbitrary continuous maps between the non-Hausdorff spaces, but to the perfect ones. Scott's insight was that computability implied a form of continuity, hence the role of the Scott topology in domains. If you restrict to perfect maps then you lose some computable maps. To put it rudely, the order (and the fact that continuity implies monotonicity and preservation of directed joins) is already naturally present in topological spaces, and the _un_natural thing to do is (1) deliberately exclude it, then (2) put it back artificially, and (3) ignore the fact that you don't quite get the same thing. The question of whether one likes the traditional notion of topological space is really focusing on the wrong things - the objects instead of the morphisms. (Actually, the traditional notion is not that attractive.) The big question is how one explains continuity, and topological spaces were set up to give an abstract definition of it. Scott's insights have elucidated continuity for us, and at the same time validated the non-Hausdorff notion of topological space (as I tried to explain in my book). And let me take this much further: Grothendieck's insights into continuity have shown that topological spaces don't go far enough. For example, he showed that it is good to extend one's ideas of continuity so that a continuous map to the "space of sets" (which doesn't exist as a topological space) is a sheaf, and a continuous map from that space is a functor preserving filtered colimits. His technique for topologization - specify the category of sheaves - goes far beyond Hausdorff spaces and brings in specialization orders that are not only non-discrete but even not orders (they are the morphism structures in categories of points). Again, the functoriality and preservation of filtered colimits is a natural and intrinsic part of this. (And we also know that when we follow Grothendieck's relativization programme along these lines then we end up using point-free spaces rather than the ordinary point-set spaces.) To conclude, I don't believe that struggles to keep topology Hausdorff are compatible with the deep insights of Scott, Grothendieck and others who have given us important new clues to the nature of continuity. Best wishes, Steve. On Wed, 07 Jul 2010 06:35:07 -0700, Vaughan Pratt <pratt@cs.stanford.edu> wrote:
I thought the point of the Lawson topology was to show the opposite: that the benefits of the Scott topology could be had without having to broaden topology beyond Hausdorff.
But if one were to so broaden it, wouldn't it be more natural to do so a la Nachbin and Priestly, with topologized posets?
But if you really like the traditional notion of a topological space in all its generality, why insist on the closure conditions on open sets when we know that dropping them gives a category with reasonable properties, namely extensional Chu(Set,2), further improved to a very nice category by dropping extensionality, and generalizable to Chu(Set,K) and yet further to Chu(V,k)?
Vaughan Pratt
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