Re: non-Hausdoff topology

Re: non-Hausdoff topology

[Vaughan Pratt]

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

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Re: non-Hausdoff topology

[Steve Vickers]

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
>



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non-Hausdoff topology

[Paul Taylor]

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


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Re: non-Hausdoff topology

[Vaughan Pratt]

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
>

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Re: non-Hausdoff topology

[Steven Vickers]

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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Re: non-Hausdoff topology

[Robert J. MacG. Dawson]

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


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Re: non-Hausdoff topology

[Martin Escardo]

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.


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Re: non-Hausdoff topology

[Jeff Egger]

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
>



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Re: non-Hausdoff topology

[Steve Vickers]

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
>


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Re: non-Hausdoff topology

[Michael Barr]

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
>


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Re: non-Hausdoff topology

[Erik Palmgren]

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



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