On metric types that are definable in an o-minimal structure
نویسنده
چکیده
In this paper we study the metric spaces that are definable in a polynomially bounded ominimal structure. We prove that the family of metric spaces definable in a given polynomially bounded o-minimal structure is characterized by the valuation field Λ of the structure. In the last section we prove that the cardinality of this family is that of Λ. In particular these two results answer a conjecture given in [SS] about the countability of the metric types of analytic germs. The proof is a mixture of geometry and
منابع مشابه
On the Topology of Metric Spaces Definable in o-minimal expansions of fields
We study the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set equipped with its euclidean topology. This implies that a separable metric space which is definable in an o-minimal expansion of the real field is definably h...
متن کاملDefinable functions continuous on curves in o-minimal structures
We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal structure such that, for any bounded definable function, there exists a definable closed set containing an initial segment of the curve on which the function is continuous. This question is translated into one on types: What are the conditions on an n-type such that, for any bounded definable function, there ...
متن کاملDefinable Structures in O-minimal Theories: One Dimensional Types
Let N be a structure definable in an o-minimal structureM and p ∈ SN (N), a complete N -1-type. If dimM(p) = 1 then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N -definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to...
متن کاملVanishing Homology Guillaume
In this paper we introduce a new homology theory devoted to the study of families such as semi-algebraic or subanalytic families and in general to any family definable in an o-minimal structure (such as Denjoy-Carleman definable or ln − exp definable sets). The idea is to study the cycles which are vanishing when we approach a special fiber. This also enables us to derive local metric invariant...
متن کاملA pathological o-minimal quotient
We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure M whose elementary diagram does not eliminate imaginaries. We also give a positive answer to a related question, showing that any imaginary in a...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- J. Symb. Log.
دوره 73 شماره
صفحات -
تاریخ انتشار 2008