Re: Reference requested

["Fred E.J. Linton" <fejlinton@usa.net>]

Peter May, in re the Subject: categories: Re: Reference requested, wrote

> ... I prefer `chaotic' to `indiscrete' not just because
> of the `coarse' implications of the latter, but because
> indiscrete spaces are boring, `null or banal', whereas
> chaotic categories have genuinely significant applications. ...

Be that as it may, I sought Search-engine advice regarding the use of the  
'chaotic topological space' lingo, and came up with the following 'hits',
of which only the first reflects, in an afterthought, Peter's usage,
while the others all envision something rather quite different:

1) From  http://en.wikipedia.org/wiki/Grothendieck_topology : 

The discrete and indiscrete topologies

Let C be any category. To define the discrete topology, we declare all sieves
to be covering sieves. If C has all fibered products, this is equivalent to
declaring all families to be covering families. To define the indiscrete
topology, we declare only the sieves of the form Hom(−, X) to be covering
sieves. The indiscrete topology is also known as the biggest or chaotic
topology, and it is generated by the pretopology which has only isomorphisms
for covering families. A sheaf on the indiscrete site is the same thing as a
presheaf.

Other uses of 'chaotic', having nothing to do with indiscreteness,
predominate:

2) From http://www.math.uh.edu/~hjm/pdf26%284%29/03chara.pdf ,
reflecting the content of 

ON GENERALIZED RIGIDITY
by JANUSZ J. CHARATONIK
from Houston Journal of Mathematics (© 2000 University of Houston)
Volume 26, No. 4, 2000 :

A nondegenerate topological space X is said to be:

(a) chaotic if for any two distinct points p and q of X there exists an open
neighbourhood U of p and an open neighbourhood V of q such that no open
subset of U is homeomorphic to any open subset of V ; ... [snip] ...

3)  CHAOTIC GROUP ACTIONS
www.math.zju.edu.cn/amjcu/B/200301/030108.pdf

... no chaotic group actions on any topological space with free arc. ...
... topological space which admits a chaotic group action but admits ...

4)  CHAOTIC POLYNOMIALS IN SPACES OF CONTINUOUS AND ...
personales.upv.es/almimon/Preprint%20Aron-Miralles.pdf

... show that there exist chaotic homogeneous polynomials of degree m ≥ 2.
...

So I'd imagine 'chaotic', for 'indiscrete', is best dropped, and either
'indiscrete', 'codiscrete', or 'trivial' be used instead.

NB: while it's true that the trivial (indiscrete) topology on a set X 
is initial, in the sense that, as a collection of subsets of X, it's the 
smallest that's a topology on X, the indiscrete topological space on X
is terminal among all topological spaces on X and mappings that restrict
to the identity on X; the trivial (indiscrete) pre-order on X is likewise
terminal, in the sense that, as a subset of X x X, it's the largest.

A connected pre-ordered groupoid (i.e., indiscrete category), being
equivalent to the terminal category 1, has the property that, for each
category X, it admits exactly one isomorphism class of functor from X,
but while that may make it 2-terminal or [( co | op ) lax-] terminal, 
I'd still probably prefer to avoid such ... umm ... terminalogy :-) .

Cheers, -- Fred

PS: I re-emphasize: of all the hits I found, only one amongst the first
two dozen -- the first cited above -- spoke of the trivial topology as 
the chaotic topology; ALL the others used 'chaotic' in some other way, 
DESPITE the search having been explicitly for [ chaotic topological space  ]. 
And there were "about 175,000 results" all told :-) . -- F.

PPS: Typos? Perhaps; please forgive, I couldn't spiel-chuck. -- F.



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