The notion of o-minimality makes sense without the density requirement, but I probably put it in because it was put in by the source or sources I looked at. Speaking of Steinhorn, I see here on page 11 that he also has the density requirement, so the situation seems a little weird, doesn’t it?

Regarding “without endpoints”: true, some of the sources omit that, but I also doubt that I made it up simply because I would not have thought to do that. :-) I may have been reading from van den Dries’s Tame Topology and O-minimal structures. As you probably know, the theory of dense linear orderings without endpoints is a complete theory, has quantifier elimination, etc., and I’d guess some authors may have wanted to exploit this and related facts to help develop their accounts.

As you point out, many of the examples people seem interested in are structures that expand a structure of real closed field (certainly the case with the van den Dries book). In any case, I hope you won’t mind holding off on further edits for the time being while we look into this. Thanks for linking to Conant’s site; I’ll have a look.

]]>I need to get my hands on the book by van den Dries, but these notes by Starchenko (who is expert) has the dense linear ordering definition, as does Wkipedia (which refers the reader to Model Theory by David Marker, another expert).

I have rolled back to v13.

]]>O-minimal structures need not be dense, ex: (Z, <) is o-minimal

]]>Added pointers to recent work in Physics that proposes tame topology for finiteness in QFT.

]]>fixed the link to *dense linear order*

Replaced “Diophantine equations” with “number theory” and “arithmetic geometry”, and added links.

]]>Added relation to Diophantine equations.

]]>Added reference

- Reid Barton, Johan Commelin,
*Model categories for o-minimal geometry*(arXiv:2108.11952)

M. J. Edmundo, N. J. Peatfield,

*O-minimal Čech cohomology*, (2006) pdfRicardo Bianconi, Rodrigo Figueiredo,

*O-minimal de Rham cohomology*, arxiv/1904.05485

Related new entry tame topology.

]]>More references at o-minimal structure, and redirect o-minimal theory.

]]>Cool, thanks!

]]>I started some short articles on o-minimal structure and structure (model theory).

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