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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]