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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