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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 5th 2011

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

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeAug 10th 2011

    Cool, thanks!

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeMay 29th 2012

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

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeNov 9th 2012

    Related new entry tame topology.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeAug 9th 2021
    • M. J. Edmundo, N. J. Peatfield, O-minimal Čech cohomology, (2006) pdf

    • Ricardo Bianconi, Rodrigo Figueiredo, O-minimal de Rham cohomology, arxiv/1904.05485

    diff, v9, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 27th 2021

    Added reference

    diff, v10, current

    • CommentRowNumber7.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 5th 2022

    Added relation to Diophantine equations.

    diff, v11, current

    • CommentRowNumber8.
    • CommentAuthorAnton Hilado
    • CommentTimeJul 5th 2022

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

    diff, v11, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 5th 2022

    fixed the link to dense linear order

    diff, v12, current

    • CommentRowNumber10.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 25th 2023

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

    diff, v13, current

    • CommentRowNumber11.
    • CommentAuthorscott
    • CommentTimeAug 4th 2023

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

    diff, v14, current

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 4th 2023

    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.

    • CommentRowNumber13.
    • CommentAuthorScottB
    • CommentTimeAug 5th 2023
    Hmmm, maybe different authors have different notions, or perhaps densely ordered structures are more interesting to Starchenko, but (maintained by Gabriel Conant) which in turn cites Pillay and Steinhorn's Definable Sets in Ordered Structures (which as far as I'm aware is the original paper introducing o-minimality) only requires linear orderings. Scrolling through Marker's textbook he never seems to specify if it needs to be dense, although the only examples provided are dense. Either way, Starchenko's notes do not require there to be no endpoints.
    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 5th 2023

    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.