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TOPOLOGICALLY 1-BASED T-MINIMAL STRUCTURES
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- Journal:
- Journal of the Institute of Mathematics of Jussieu , First View
- Published online by Cambridge University Press:
- 26 October 2026, pp. 1-76
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We prove group-existence and structure theorems in a general setting of tame topological theories. More precisely, we identify a linear/non-linear dividing line – called topological 1-basedness – among the class of t-minimal theories with the independent neighbourhood property. This is a wide class including all visceral theories, as well as all dense weakly o-minimal and C-minimal theories (even those where exchange fails).
Now assume $\mathcal M$
is highly saturated and t-minimal with the independent neighbourhood property. We show that if $\mathcal M$
is non-trivial and topologically 1-based, it admits a type-definable abelian group $(G,+)$
with G an open subseteq of M. Moreover, we can ensure that G is a topological group with the subspace topology inherited from M; and in this case, we show that the induced structure on G satisfies an appropriate topological analogue of the Hrushovski-Pillay classification of 1-based stable groups.
