O-Minimal Structures

• O-minimal structures are mathematical frameworks defined by expansions of dense linear orders where every definable one-variable set is a finite union of points and open intervals.
• They enable robust cell decomposition and dimension theory, providing clear stratifications and topological regularity for definable sets.
• Applications extend to definable measure, duality theories, and classifications of groups and orders, enriching tools in tame model theory and real algebraic geometry.

An o-minimal structure is a model-theoretic expansion of a dense linear order without endpoints such that every definable subset of the universe in one variable is a finite union of points and open intervals. This property, known as o-minimality, serves as a foundational paradigm for tameness in definable sets and functions, enabling a powerful intersection of methods from real algebraic geometry, model theory, topology, and combinatorics. O-minimal structures admit robust cell decomposition and dimension theory, support analogues of classical topology such as definable compactness, and provide a context for duality, measure, and group theory in “tame” categories. Recent advances have sharpened this landscape via complexity-controlled filtrations, metric and quasi-analytic generalizations, and links with NIP, dp-minimal, and continuous logic settings.

1. Definition and Core Properties

Let $\mathcal M = (M,<,\ldots)$ be an expansion of a dense linear order without endpoints. $\mathcal M$ is o-minimal if for every $A \subset M$ definable (possibly with parameters), $A$ is a finite union of points and open intervals. Equivalently, for every $k \ge 1$, every definable subset of $M^k$ admits a cell decomposition: a finite partition into definable sets (cells), each homeomorphic to an open box by coordinate projection (Guerrero, 2021).

Key features:

• Cell decomposition ensures all definable sets have a canonical stratification by dimension, and the closures of cells decompose into unions of lower-dimensional cells.
• Dimension theory is robust: for $X\subseteq M^n$, the dimension is the maximal $d$ such that some projection to $M^d$ has nonempty interior. Fiber and addition formulas hold (Fujita, 2020).
• Tameness manifests as strong regularity in topology, measure, and algebraic structure: finiteness in complexity, local triviality of definable decompositions, boundedness of complexity under logical operations.

Typical examples:

Structure	Description
$(\mathbb R,<,+,\times)$	Real field
$(\mathbb R,<,+,\times,\exp)$	Real exponential field (Wilkie’s structure)
$(\mathbb R,<,+,\{\!f_i\!:\!i\in I\})$	Expansions by restricted analytic or Pfaffian functions
Locally o-minimal expansions with DCTC	Dense linear orders with “local” finiteness

2. Definable Topologies and Compactness

A definable topological space in an o-minimal structure is a pair $(X,\tau)$, with $X\subseteq M^n$ definable and $\tau$ admitting a uniformly definable basis $\{U(u)\subseteq X: u\in\Omega\}$ for $\Omega\subseteq M^k$ definable (Guerrero, 2021). Within such spaces, notions of compactness are natural and admit robust characterizations:

• Definably compact: Every downward-directed definable family of nonempty $\tau$-closed sets has nonempty intersection.
• Type-compact: Every definable type on $X$ has a limit.
• Curve-compact: Every definable curve $\gamma:(a,b)\rightarrow X$ converges in $\tau$ at both ends.

These properties coincide under mild auxiliary conditions (e.g., Hausdorffness or definable choice), and generalize the classical closed + bounded compactness characterization to arbitrary o-minimal definable topologies.

Specialization/Transversal Characterization

The following are equivalent for a definable topological space $(X, \tau)$:

1. $(X,\tau)$ is definably compact.
2. $(X,\tau)$ is type-compact.
3. Every definable family of $\tau$-closed sets extending to a definable type has nonempty intersection.
4. Every consistent definable family of $\tau$-closed sets admits a finite transversal.
5. For every definable family $C$ of nonempty $\tau$-closed sets with the $(p, q)$-property and $q > \dim\bigcup C$, there is a finite transversal.
6. The same as (5) where $q > \mathrm{VC}\text{-}\mathrm{codensity}(C)$.

When $\tau$ is Hausdorff or $\mathcal M$ has definable choice, these are also equivalent to curve-compactness (Guerrero, 2021).

This equivalence yields o-minimal analogues of Helly-type intersection theorems and underpins a range of “tame” compactness results for definable sets.

3. Cell Decomposition, Dimension, and Tame Topology

Central to o-minimal structures is the cell decomposition theorem: any definable set can be partitioned into finitely many cells—definable, locally manifold pieces homeomorphic to open boxes—stratified by dimension (Shiota, 2010). Immediate consequences:

• Dimension theory: $\dim(X)$ for definable $X\subset M^n$ coincides with classical manifold dimension on cells, is preserved under projections, products, unions, and preimages, and satisfies the addition property: if $f:X\to Y$ is definable with constant fiber dimension $d$, then $\dim(X) = \dim(Y) + d$ (Fujita, 2020).
• Triangulation and PL topology: Any compact definable set is definably homeomorphic to a polyhedron; the o-minimal Hauptvermutung asserts the uniqueness of such PL structures up to homeomorphism. PL-topological constructions, microbundles, cobordism, and smoothing arguments all carry over to the o-minimal context (with fields or real closed coefficients) (Shiota, 2010).
• Sharp cell decomposition: In sharply o-minimal structures, cell decomposition is augmented with explicit complexity control (format/degree), bounding the number, format, and degree of the cells, and producing strong uniformity results and explicit bounds on Betti numbers (Binyamini et al., 2022).

For locally o-minimal and definably complete expansions, global cell decomposition may fail; instead, there is a robust theory of quasi-special submanifold decomposition, with analogous dimension and topological properties (Fujita, 2020, Schoutens, 2011).

4. Measures, Duality, and Sheaf-Theoretic Structures

The robust geometric nature of o-minimal structures supports a theory of measure, cohomology, and duality with properties analogous to classical settings:

• Hausdorff measure: One can define an o-minimal version of $e$-dimensional Hausdorff measure, $\mathcal H^e$, by integrating Jacobians over the coordinate charts of basic rectifiable pieces (e.g., $C^1$-graphs), extending classical geometric measure theory to the definable context. Core integral-geometric identities—such as the Cauchy–Crofton and co-area formulae—hold in this setting, enabling applications to isoperimetry and point counting (Fornasiero et al., 2010).
• Sheaf cohomology and duality: Verdier and Poincaré duality theorems extend to the o-minimal context, with definably compact supports, yielding Eilenberg–Steenrod type axioms for cohomology, orientation sheaves, and natural dualizing complexes. These formal structures enable computation and duality for definable groups and spaces, and suggest avenues for microlocal and Riemann–Hilbert theory in tame settings (Edmundo et al., 2010).

5. Definable Groups, Linear Orders, and Classification

Groups and definable linear orders in o-minimal structures exhibit strict classification results that leverage cell and dimension theory:

• Definable groups: In any “tame” expansion of an o-minimal structure, groups which are “strongly large”—their dimension matches that of their closure—are already interpretable in the reduct, i.e., there are no genuinely “new” large definable groups in tame expansions. The group chunk theorem formalizes the extension of partial group laws to full groups in this context (Eleftheriou, 2018).
• Linear orders: Any definable linear order of dimension $n$ in an o-minimal structure (expanding a group) embeds definably into $(M^{2n+1}, <_{\mathrm{lex}})$ with controlled projections; fields admit even sharper embeddings. This supports a fine classification of definable orders and bridges model theory and order theory (Ramakrishnan, 2010).

6. Extensions: Weak o-Minimality, O-Minimalism, Metrics, and Quasianalytic Expansions

The o-minimal paradigm generalizes in several directions:

• O-minimalism and DCTC: O-minimalistic structures satisfy all first-order consequences (completeness, local finiteness, dimension theory), synthetically captured by the Definable Completeness and Type Completeness (DCTC) scheme, and—optionally—the discrete pigeonhole principle (DPP) (Schoutens, 2011). Tameness and quasi-cell decomposition replace finitary cell decomposition, supporting an analysis of ultraproducts and invariants such as Grothendieck rings and ultraproduct Euler characteristics.
• Locally o-minimal/Definably complete structures: These admit versions of dimension, monotonicity, and decomposition, controlled under the discrete-projection property, yet may lack global cell decomposition (Fujita, 2020).
• Metric and continuous logic: O-minimality adapts to the metric context via the notion of regulated (or strongly regulated) functions, leading to o-minimal or weakly o-minimal metric structures, definable completeness, and distality with analogues of quantifier elimination and density (e.g., real closed metric valued fields, ultrametric cyclic orders) (Anderson et al., 13 Oct 2025).
• Quasianalytic/weak smooth expansions: The scope of o-minimality may be extended to certain quasi-analytic or Hardy-field functional frameworks where smooth cell decomposition fails but cell decomposition persists in a weakened geometric sense (e.g., weakly smooth germs); this generates new o-minimal structures containing nowhere-smooth definable functions (Guénet, 12 Mar 2025).

7. Definable Topologies, Interpretability, and Path-Connectedness

Interpretables and quotients in o-minimal structures, though not always susceptible to elimination of imaginaries, nevertheless admit tame topologies (Hausdorff, locally Euclidean, finite-dimensional) and path-connectedness properties. For o-minimal expansions of $(\mathbb R,<)$, path components of definable sets are again definable; any expansion with definable path components is either o-minimal or encodes full arithmetic, creating a dichotomy between tameness and undecidability (Dolich et al., 2020, Johnson, 2019).

Table: Structural Theorems and Applications

Area	Key Theorem/Property	Reference
Cell decomposition	All definable sets decompose finitely	(Shiota, 2010)
Definable compactness	Equivalences via types/transversals	(Guerrero, 2021)
Hausdorff measure	Existence, Crofton/co-area formulas	(Fornasiero et al., 2010)
Duality	Verdier, Poincaré duality	(Edmundo et al., 2010)
Definable groups	No new large groups in tame expansions	(Eleftheriou, 2018)
Linear orders	Embeddings into lexicographic orders	(Ramakrishnan, 2010)
O-minimalistic/DCTC	Full first-order abstraction	(Schoutens, 2011)
Quasianalytic structures	Weakly smooth germs yield o-minimality	(Guénet, 12 Mar 2025)
Metric o-minimality	Regulated functions, distality	(Anderson et al., 13 Oct 2025)
Sharp o-minimality	Cell/Betti complexity control	(Binyamini et al., 2022)

8. Open Questions and Future Directions

Research continues on characterizing definable compactness by first-order schemes, understanding parameter-definability of compactness in families, microlocalization of sheaf theory, and extensions to broader tame contexts (e.g., analytic, metric, and non-archimedean settings) (Guerrero, 2021, Edmundo et al., 2010). The refinement of sharp complexity, further extensions of o-minimal duality or measure theory, and classification of expansion hierarchies remain active areas with significant structural and arithmetic implications.

O-minimal structures thus provide a rich and flexible framework for understanding model-theoretic tame geometry, linking deep topological, combinatorial, and analytic properties of definable sets across diverse mathematical landscapes.