If $X$ is a set, some sources* refer to the topology $\{\emptyset,X \}$ as the chaotic topology. (I've also seen it called the trivial, codiscrete, and indiscrete topology.) What is the origin of and motivation for this term?

The term discrete topology makes sense to me because the restriction of the Euclidean topology on $\mathbb R$ to a set of discrete points results in the discrete topology. But I can see no such explanation for the term chaotic topology.

* [1] [2] [3] [4]

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    $\begingroup$ Perhaps a motivation was that in such a topology there is no locality, no way to distinguish one location (nbhd) from another. But I've never seen this term before. Another term for it is anti-discrete. I like "indiscrete", but only because of its non-mathematical social nuances in English. $\endgroup$ – DanielWainfleet Jan 6 at 7:44
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    $\begingroup$ It seems that there is some connection to algebraic geometry (perhaps you should add the tag). See en.wikipedia.org/wiki/Grothendieck_topology. Also see M. Artin, A. Grothendieck, J-L. Verdier, eds. (1972), SGA4, LNM 269-270-305, Berlin; New York: Springer-Verlag: Exposé IV, 2.6 . I found this reference in arxiv.org/pdf/1405.4527.pdf. $\endgroup$ – Paul Frost Jan 6 at 17:36
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    $\begingroup$ The lecturer of the first topology course i ever took used the word chaotic for the indiscrete topology and said this comes from a bible quote: something along the lines of "at the beginning the world was chaotic and empty". Not sure whether that is true but that is at least what he said. $\endgroup$ – ThorWittich Jan 7 at 0:35
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    $\begingroup$ @ThorWittich From the New International Version of the Bible (Genesis 1:1-2): "In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters." See en.wikipedia.org/wiki/Genesis_1:2. The orginal Hebrew phrase is "tohu wabohu" which some people freely translate by "chaos". See en.wikipedia.org/wiki/Chaos_(cosmogony)#Greco-Roman_tradition . $\endgroup$ – Paul Frost Jan 7 at 10:36
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    $\begingroup$ @ThorWittich Sounds reasonable. In fact, a space with the indiscrete topology does not satisfy any separation properties. Now the next two verses of Genesis say "And God said, “Let there be light,” and there was light. God saw that the light was good, and he separated the light from the darkness." $\endgroup$ – Paul Frost Jan 7 at 10:48

I don't know if this is the original motivation: When $\leq$ is a partial order on a set $S$, it induces a topology called the order topology, which is the one generated by the open rays $\{ c \in S : c < a \}, \{ c \in S : c > a \}$. The order topology is chaotic precisely if the order is trivial, i.e. if every element is comparable only with itself.

This may explain "chaotic", as the antonym of "ordered".

It seems that a similar point of view was already taken by Felix Hausdorff in Grundzüge der Mengenlehre (1922). He talks about ordered sets in chapters 4–6, and in chapter 7 introduces a notion of a topological space. The introduction to chapter 7 (pp. 209–211) views an order as an example to give additional structure to a set, and goes on to explore generalizations of the concept of an order, and arrives at the notion of neighborhood. Hausdorff remarks that an order can be defined from a suitable system of neighborhoods:

Hier wird also eine Menge $M$ unter dem Gesichtspunkt einer Zuordnung zwischen Elementen und Teilmengen betrachtet; wir haben übrigens gezeigt (Kap. IV, § 1), daß man auch die Ordnung einer Menge durch ein passendes System von Teilmengen definieren kann.

I do not know who first used the word chaotic (or French chaotique, German chaotisch) in the context of topology. It does not appear in Hausdorff's book.

  • $\begingroup$ In fact it would be interesting to know who first introduced the phrase. $\endgroup$ – Paul Frost Jan 8 at 14:41

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