O-minimality

A structure $$(M,<,\ldots)$$ is said to be o-minimal if every subset $$X \subset M^1$$ definable with parameters from $$M$$ can be written as a finite union of points and intervals, i.e., as a boolean combination of sets of the form $$\{x \in M : x \le a\}$$ and $$\{x \in M : x \ge a\}$$. Note that this is an assertion about subsets of $$M^1$$, not definable sets in higher dimensions.

This notion is analogous to minimality. In minimality, one assumes that the definable (one-dimensional) sets are quantifier-free definable using nothing but equality. Here, one assumes that the (one-dimensional) sets are quantifier-free definable using nothing but the ordering.

A theory $$T$$ with a predicate $$<$$ is said to be o-minimal if every model of $$T$$ is o-minimal. Unlike the case of minimality vs. strong minimality, there is no notion of strong o-minimality. It turns out that any elementary extension of an o-minimal structure is o-minimal. Consequently, the true theory of any o-minimal structure is an o-minimal theory. The proof of this is rather non-trivial, and uses the cell decomposition result for o-minimal theories.

Often one restricts to the class of o-minimal structures/theories in which the ordering $$(M,<)$$ is dense, i.e., a model of DLO. Most o-minimal theories of interest have this property, and many proofs can be simplified with this assumption.

Examples
Some relatively elementary examples:


 * DLO, the theory of dense linear orders. This is the true theory of $$(\mathbb{Q},<)$$.
 * RCF, the theory of real closed fields. This is the true theory of $$\mathbb{R}$$ as an ordered field.
 * DOAG or ODAG, the theory of divisible ordered abelian groups. This is the true theory of $$(\mathbb{R},+,<)$$.

By hard theorems of Alex Wilkie and other people, certain expansions of the ordered field $$\mathbb{R}$$ are known to be o-minimal.


 * The structure $$\mathbb{R}_{exp} := (\mathbb{R},+,\cdot,\exp)$$ was proven to be o-minimal by Alex Wilkie. This structure consists of the ordered field $$\mathbb{R}$$ expanded by adding in a predicate for the exponentiation map. This example is somewhat surprising, given that we lack a recursive axiomatization of this structure.
 * The structure $$\mathbb{R}_{an}$$, consisting of the ordered field $$\mathbb{R}$$ with restricted analytic functions, is o-minimal. For each real-analytic function $$f$$ on an open neighborhood of $$[0,1]^n$$, one adds a function symbol for $$f$$ restricted to $$[0,1]$$. This does not subsume $$\mathbb{R}_{exp}$$, since $$\exp$$ turns out to not be definable in $$\mathbb{R}_{an}$$. In fact the o-minimality of $$\mathbb{R}_{an}$$ is a more basic result. It is essentially Gabrielov's theorem.
 * More generally, $$\mathbb{R}_{an,exp}$$ is o-minimal. This is the expansion of $$\mathbb{R}$$ obtained by adding in both the exponential map and the restricted analytic functions.
 * More generally, one can add all Pfaffian functions. The most general result in this direction is due to Speissegger, maybe.

Properties
O-minimal theories are NIP, but never stable or simple, as they have the order property. O-minimal theories are also superrosy, of finite rank.

In any o-minimal theory, definable closure and algebraic closure agree (on account of the ordering), and these operations define a pregeometry on the home sort. This yields a well-defined notion of dimension of definable sets.

Not all o-minimal theories eliminate imaginaries, even after naming all parameters from a model. However, o-minimal expansions of RCF always eliminate imaginaries, and in fact have definable choice (which includes definable Skolem functions). The same holds for o-minimal expansions of DOAG after naming at least one non-zero element.

Definable functions and definable sets have many nice structural properties. For simplicity assume that the order is dense. Then one has the following results:


 * Every definable function $$f : M^1 \to M^n$$ is piecewise continuous: the domain of $$f$$ can be written as a finite union of intervals, such that on each interval, $$f$$ is continuous. If $$n = 1$$, then one can also arrange that on each interval, $$f$$ is either constant, or strictly increasing, or strictly decreasing.
 * Every definable subset of $$M^n$$ has finitely many definably connected components. In the presence of definable Skolem functions, each piece is definably path-connected.
 * More precisely, every definable subset $$X \subset M^n$$ has a cell-decomposition: it can be written as disjoint union of sets that are "cells" in a certain sense. Each cell is definably connected, and in the case of o-minimal expansions of RCF, is definably homeomorphic to a ball.
 * If $$f : M^n \to M^m$$ is a definable function, then the domain of $$f$$ has a cell decomposition such that the restriction of $$f$$ to each cell is continuous.
 * If $$X \subset M^n$$, the topological closure $$\overline{X}$$ of $$X$$ has dimension no bigger than $$X$$, and the frontier $$\overline{X} \setminus X$$ has strictly lower dimension than $$X$$.

Many of the topological pathologies that are common in pointset topology and real analysis don't occur when working with definable sets in o-minimal expansions of the reals. For example, every definable set is locally path-connected, every connected component is path connected, every set without interior is nowhere dense, and every definable set is homotopy equivalent to a finite simplicial complex. Moreover, every continuous definable function is piecewise differentiable, and in fact piecewise $$C^k$$ for every $$k < \infty$$. One also knows that if $$f$$ and $$g$$ are two definable functions $$\mathbb{R} \to \mathbb{R}$$, then $$f$$ and $$g$$ are asymptotically comparable. Limits always exist (possibly taking values $$\pm \infty$$).

These results apply in particular to, e.g., the structure $$(\mathbb{R},+,\cdot,\exp)$$. In sharp contrast, the definable sets in $$(\mathbb{R},+,\cdot,\sin)$$ are exactly the sets in the projective hierarchy, so e.g. there are definable sets which are not Borel.

O-minimal trichotomy
Some analog of the Zilber trichotomy holds in the o-minimal setting.

Applications
Real algebraic geometry, Pila-Wilkie…