Re: Reference requested

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

I recently reported that

> ... I sought Search-engine advice regarding the use of the 
> 'chaotic topological space' lingo, and came up with ...

... surprisingly little. Early, early this morning I tried that again,
but enclosing the search string in double-quotation marks. Ah, now I
found two other references, worth citing, perhaps:

1) Volker Runde's 2005 Springer Universitext, isbn=038725790X, 
A Taste of Topology, includes a passage (page 72), beginning 

"Let (X,TX) be a chaotic topological space (ie, TX = {∅,X}), 
let (Y,TY) be a Hausdorff space, and let f: X → Y be continuous"
  
and deducing such f must be constant; links (to Google Books and a PDF):

[long url omitted by moderator],

ftp://210.45.114.81/math/2007_07_06/Universitext/V.Runde%20A%20Taste%20of%20Topology.pdf


(no hint, though, how 'standard' Runde thought his use of "chaotic" here was
:-{ ); and

2) Mat{´ı}as Menni's 2000 Edinburgh PhD thesis {Exact Completions and
Toposes} makes
multiple mention of chaotic structures, with frequent citations of Bill
Lawvere's 
interest in such things. All of Chapter 7 is about "Chaotic Situations", with
Section 
7.1 focused in particular on "Chaotic Objects"; and Section 8.4 returns to
"Chaotic Situations". A hint of the flavor is given in the Introduction
already:

"In 1999, Longley introduced a typed version of the notion of a partial 
combinatory algebra in [68] and described how to build a category of
assemblies 
Ass(A) over a [sic] such a structure A. Shortly after, Lietz and Streicher
showed 
that the ex/reg completion of Ass(A) is a topos if and only if the typed
structure 
A is equivalent, in a suitable sense, to an untyped structure. Their proof
uses 
the notion of a generic mono (a mono τ such that every other mono arises as a

pullback of τ along a not necessarily unique map) and of the constant-objects

embedding of Set into the category Ass(A) which they see as an inclusion of 
codiscrete objects. Related to this, it should be mentioned that Lawvere had 
already advocated for a conceptual use of codiscrete or chaotic objects in 
other areas of mathematics (see for example [59, 55, 61, 63])."

No surprise, then, to see the right adjoint ∇ to the forgetful functor Set
-> Top
described (p. 23) as follows: " ... the functor ∇: Set -> Top assigns to
each set 
S the “chaotic” topological space with underlying set S and, as open sets,

only S itself and the empty set."

Cf.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.22.9817&rep=rep1&type=pdf

---

BTW, those four Lawvere references are these:

[55] F. W. Lawvere. Toposes generated by codiscrete objects in combinatorial
topology and functional analysis. Notes for colloquium lectures given at
North Ryde, New South Wales, Australia on April 18, 1989 and at Madison
USA, on December 1, 1989.

[59] F. W. Lawvere. Categories of spaces may not be generalized spaces as
exemplified by directed graphs. Revista colombiana de matem´aticas,
20:179–
186, 1986.

[61] F. W. Lawvere. Some thoughts on the future of category theory. In
Proceed-
ings of Category Theory 1990, Como, Italy, volume 1488 of Lecture notes
in mathematics, pages 1–13. Springer-Verlag, 1991.

[63] F. W. Lawvere. Unit and identity of opposites in calculus and physics.
Applied categorical structures, 4:167–174, 1996.

---

All told, eight hits, all either these two, or references to them, or
search-database errors :-) . Not very heavy evidence in favor of "chaotic".

So: cheers -- and back to [co-|in-]discrete, I fear :-) , -- Fred 



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