Re: Terminology for point-free topology?

[Pedro Resende <pedro.m.a.resende@tecnico.ulisboa.pt>]

In addition to all the deeper reasons, `pointless’ can be taken to be derogatory, so preferably it should be used only when in tongue-in-cheek mode. At least that’s what I tell my students — just as I ask them not to say `abstract nonsense’ too enthusiastically… :)

Pedro

> On Jan 21, 2023, at 7:42 PM, ptj@maths.cam.ac.uk wrote:
> 
> I was wondering how long it would be before someone in this thread
> referred to my `point of pointless topology' paper! Perhaps not so many
> people know that the title was a conscious echo of an earlier paper
> by Mike Barr called `The point of the empty set', which began with the
> words (I quote from memory) `The point is, there isn't any point there;
> that's exactly the point'.
> 
> As Steve says, to fit that title I had to use the word `pointless', but
> on the whole I prefer `pointfree'; it carries the implication that you
> are free to work without points or to use them (in a generalized sense),
> as you prefer.
> 
> Peter Johnstone
> 
> On Jan 21 2023, Steven Vickers wrote:
> 
>> Dear David,
>> 
>> Yes, and it's an excellent paper with a witty title for which only
>> "pointless" would do.
>> 
>> I particularly like what Peter said when explaining the significant
>> difference in the absence of choice (such as in toposes of sheaves), and
>> that "usually it is locales, not spaces, which provide the right context
>> in which to do topology".
>> 
>> He went on to say,
>> 
>> "This is the point which ... Andre Joyal began to hammer home in the
>> early 1970s; I can well remember how, at the time, his insistence that
>> locales were the real stuff of topology, and spaces were merely figments
>> of the classical mathematician's imagination, seemed (to me, and I
>> suspect to others) like unmotivated fanaticism. I have learned better
>> since then."
>> 
>> This is all part of the argument for using a reformed topology, but there
>> is nothing particular there about the pointwise style of reasoning for
>> it. Hence we are still left with the question of how to reference the two
>> concepts, the reformed topology and the reasoning without points.
>> 
>> Would you call Ng's paper with me pointless? Points are everywhere in it.
>> (Of course, there's the separate issue of whether it was pointless in the
>> sense of not worth the trouble. But an important feature of the style is
>> that it forces you to be careful to distinguish between Dedekind reals
>> and 1-sided (lower or upper) reals, and in Ng's thesis this uncovered
>> unexpected roles of 1-sided reals in the account of Ostrowski's Theorem
>> and the Berkovich spectrum. So there is a bit of payoff.)
>> 
>> Best wishes,
>> 
>> Steve.
>> 

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