Non-Hausdorff topologies, in particular the Scott topology, have been one of the most important features of mathematics applied to computer science over the past forty years. Surely it is now time for this material to be included in the standard undergraduate curriculum for general topology in pure mathematics degree programmes. I wonder whether "categories" reader have some comments on their experience of trying to do this? I am thinking of the possible reactions from both students and colleagues. Paul Taylor [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 On 7/7/2010 1:31 AM, Paul Taylor wrote:
Non-Hausdorff topologies, in particular the Scott topology, have been one of the most important features of mathematics applied to computer science over the past forty years.
Surely it is now time for this material to be included in the standard undergraduate curriculum for general topology in pure mathematics degree programmes.
I wonder whether "categories" reader have some comments on their experience of trying to do this? I am thinking of the possible reactions from both students and colleagues.
Paul Taylor
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 7/7/2010 5:31 AM, Paul Taylor wrote:
Non-Hausdorff topologies, in particular the Scott topology, have been one of the most important features of mathematics applied to computer science over the past forty years.
Surely it is now time for this material to be included in the standard undergraduate curriculum for general topology in pure mathematics degree programmes.
Dear Paul et al: I certainly learned about non-Hausdorff topologies in the topology course I took as an undergraduate from Michael Edelstein at Dalhousie (using Kelley's "General Topology" as a text). The Zariski topology also appeared in a couple courses, and various instructors recommended the book "Counterexamples in Topology" by Steen and Seebach, which gives a fairly good "tour of the zoo". Nonetheless, thirty year later, I would certainly accept that a modern treatment of the topic would have a somewhat different focus, for precisely the reasons that you give in your first paragraph. -Robert [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Paul Taylor wrote:
Non-Hausdorff topologies, in particular the Scott topology, have been one of the most important features of mathematics applied to computer science over the past forty years.
Surely it is now time for this material to be included in the standard undergraduate curriculum for general topology in pure mathematics degree programmes.
I wonder whether "categories" reader have some comments on their experience of trying to do this? I am thinking of the possible reactions from both students and colleagues.
Ok, here is an experience (about 3 years ago, I think). One day I went for lunch on my own at the university's Staff House, and I sat at a table in which the only other person turned out to be an analyst in the maths department, at retirement age. After I asked about his work, he asked about mine, and I said I was in computer science, and that parts of my work involved the use of topology in understanding computation. So far so good, and I had an attentive and inquisitive listener for probably more than 1/2 hour, at which point he queried more about the nature of the spaces one comes up with in this field. The first thing I answered was that often they were not Hausdorff. And that was also the last, because he looked in amazement and disbelief, said that then these were not really topological spaces, checked his watch, said something incomprehensible, and left without further ado. MHE. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi Paul, In my experience, there is tremendous resistance (both from students and professors) to the idea that Topology can be anything other than a handmaiden to Analysis and Analytic Geometry. Even applications to Algebraic Geometry are viewed with deep suspicion, and sometimes even brushed aside as "not real Topology". So what is "real Topology"? There is a precise theorem to the effect that completely regular T_0 spaces (a.k.a., Tychonov spaces) form the maximum subcategory of Top which can be of interest to (conventional) analysts. This, as I'm sure you're aware, is the essential image of the forgetful functor Unif --> Top. So perhaps "real Topology" is as much about uniform spaces as it is about topological spaces? (Note also the prevalence of topological groups in analysis, and the equivalence between (separated) uniform groups and (T_0) topological groups.) Given that every topological space is quasi-uniformisable, it seems that the problem of motivating non-Tychonov (and, in particular, non-Hausdorff) spaces is actually equivalent to that of motivating non-symmetric metric spaces! Cheers, Jeff. ----- Original Message ----
From: Paul Taylor <pt10@PaulTaylor.EU> To: categories@mta.ca Sent: Wed, July 7, 2010 9:31:15 AM Subject: categories: non-Hausdoff topology
Non-Hausdorff topologies, in particular the Scott topology, have been one of the most important features of mathematics applied to computer science over the past forty years.
Surely it is now time for this material to be included in the standard undergraduate curriculum for general topology in pure mathematics degree programmes.
I wonder whether "categories" reader have some comments on their experience of trying to do this? I am thinking of the possible reactions from both students and colleagues.
Paul Taylor
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Paul, At the Open University I was in the course development team for a thorough revision of the Topology course. Some of the other members started with the opinion that there was no need to go beyond metric spaces. However, it did eventually include general topology, mostly Hausdorff, and material on surfaces and their classification. I wrote a 50 page unit on "Topology and Computation" for it, including the specialization order, some finite topological spaces (at one point I had a section giving the equivalence between them and preorders), function spaces in some simple domain settings, continuity of currying and uncurrying and recursion as limit points. However, I left before the course development was finished, and the others didn't feel brave enough to include such wacky material without me there. So I thought in 2001 that it was time for this material to be included in the standard undergraduate curriculum for general topology, but failed to convince the people who might actually include it. Regards, Steve. Paul Taylor wrote:
Non-Hausdorff topologies, in particular the Scott topology, have been one of the most important features of mathematics applied to computer science over the past forty years.
Surely it is now time for this material to be included in the standard undergraduate curriculum for general topology in pure mathematics degree programmes.
I wonder whether "categories" reader have some comments on their experience of trying to do this? I am thinking of the possible reactions from both students and colleagues.
Paul Taylor
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Not just in CS, but also central to algebraic geometry: the Zariski topology is almost never hausdorff. But when topology is taught to undergraduates, it is usually for the purposes of analysis and I don't know if we could this point across. Michael On Wed, 7 Jul 2010, Paul Taylor wrote:
Non-Hausdorff topologies, in particular the Scott topology, have been one of the most important features of mathematics applied to computer science over the past forty years.
Surely it is now time for this material to be included in the standard undergraduate curriculum for general topology in pure mathematics degree programmes.
I wonder whether "categories" reader have some comments on their experience of trying to do this? I am thinking of the possible reactions from both students and colleagues.
Paul Taylor
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 7/7/2010 10:28 AM, Michael Barr wrote:
Not just in CS, but also central to algebraic geometry: the Zariski topology is almost never hausdorff. But when topology is taught to undergraduates, it is usually for the purposes of analysis and I don't know if we could this point across.
My understanding of Paul's complaint was with the passage not so much from T2 to T0 but from T1 to T0, needed for the Scott topology. The Zariski topology is always T1. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Vaughan, The Zariski topology on the prime spectrum is normally not T1. In fact Hochster showed that any spectral space (the ones that correspond to ordered Stone spaces in the Priestly duality) is homeomorphic to the prime spectrum of some ring, with its Zariski topology. In particular, this holds for Scott domains and some other classes of spaces commonly arising in denotational semantics - though it would be eccentric to treat them as spectra of rings. Referring to the Wikipedia page on Zariski topology, the "classical" definition gives a T1 topology, but the "modern" definition adds extra "generic" points corresponding to non-maximal prime ideals and they create a non-discrete specialization order, T0 but not T1. Regards, Steve. Vaughan Pratt wrote:
On 7/7/2010 10:28 AM, Michael Barr wrote:
Not just in CS, but also central to algebraic geometry: the Zariski topology is almost never hausdorff. But when topology is taught to undergraduates, it is usually for the purposes of analysis and I don't know if we could this point across.
My understanding of Paul's complaint was with the passage not so much from T2 to T0 but from T1 to T0, needed for the Scott topology. The Zariski topology is always T1.
Vaughan Pratt
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
So-called "digital" topologies are simple and illustrative examples of non-Hausdorff topologies suitable for undergraduate classes. These are finite combinatorial topologies that are used to define connectivity of images on a computer screen. See for instance Chapter 9 of these lecture notes of Christer Kiselman "Digital Geometry and Mathematical Morphology" http://www.math.uu.se/~kiselman/dgmm2004.pdf There is even a Wikipedia article http://en.wikipedia.org/wiki/Digital_topology [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (9)
-
Erik Palmgren -
Jeff Egger -
Martin Escardo -
Michael Barr -
Paul Taylor -
Robert J. MacG. Dawson -
Steve Vickers -
Steven Vickers
-
Vaughan Pratt