Re: non-Hausdoff topology

[Steven Vickers <S.J.Vickers@cs.bham.ac.uk>]

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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