# Definable one-dimensional topologies in O-minimal structures

@article{Peterzil2020DefinableOT,
title={Definable one-dimensional topologies in O-minimal structures},
author={Ya'acov Peterzil and A. Carnero Rosel},
journal={Archive for Mathematical Logic},
year={2020},
volume={59},
pages={103-125}
}
• Published 21 July 2018
• Mathematics
• Archive for Mathematical Logic
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $$\left( X,\tau \right)$$ X , τ is definably homeomorphic to an affine definable space (namely, a definable subset of $$M^{n}$$ M n with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.
3 Citations

### Affiness of topological space definable in a definably complete uniformly locally o-minimal structure of the second kind

• Mathematics
• 2021
We give a necessary and sufficient condition for a one-dimensional regular and Hausdorff topological space definable in a definably complete uniformly locally o-minimal structure of the second kind

### L O ] 1 4 N ov 2 01 9 Directed sets and topological spaces definable in o-minimal structures

• Mathematics
• 2019
We study directed sets definable in o-minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered

### Directed sets and topological spaces definable in o‐minimal structures

• Mathematics
Journal of the London Mathematical Society
• 2021
We study directed sets definable in o‐minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered

## References

SHOWING 1-8 OF 8 REFERENCES

### 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

### Definable Compactness and Definable Subgroups of o‐Minimal Groups

• Mathematics
• 1999
The paper introduces the notion of definable compactness and within the context of o‐minimal structures proves several topological properties of definably compact spaces. In particular a definable

### Tame Topology and O-minimal Structures

1. Some elementary results 2. Semialgebraic sets 3. Cell decomposition 4. Definable invariants: Dimension and Euler characteristic 5. The Vapnik-Chernovenkis property in o-minimal structures 6.

### Topologizing interpretable sets in O-minimal Structures

Let M be a structure in some language. Assume M has elimination of imaginaries. Let X be a definable set. Definable will mean “definable with parameters.” By a definable topology, we mean a definable

### On linearly Ordered Structures of finite Rank

• Mathematics
J. Math. Log.
• 2009
A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here, which include all ordered structures definable (as subsets of n-tuples of the universe) in o-minimal structures.

### First order topological structures and theories

• A. Pillay
• Mathematics, Computer Science
Journal of Symbolic Logic
• 1987
The notion of a first order topological structure is introduced, and various possible conditions on the complexity of the definable sets in such a structure are considered, drawing several consequences thereof.

### Model theory : an introduction

This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques,