Re: non-Hausdoff topology

[Jeff Egger <jeffegger@yahoo.ca>]

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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