A structure is said to be o-minimal if every subset definable with parameters from can be written as a finite union of points and intervals, i.e., as a boolean combination of sets of the form and . Note that this is an assertion about subsets of , 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 with a predicate is said to be o-minimal if every model of 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 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 .
• RCF, the theory of real closed fields. This is the true theory of as an ordered field.
• DOAG or ODAG, the theory of divisible ordered abelian groups. This is the true theory of .

By hard theorems of Alex Wilkie and other people, certain expansions of the ordered field are known to be o-minimal.

• The structure was proven to be o-minimal by Alex Wilkie. This structure consists of the ordered field 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 , consisting of the ordered field with restricted analytic functions, is o-minimal. For each real-analytic function on an open neighborhood of , one adds a function symbol for restricted to . This does not subsume , since turns out to not be definable in . In fact the o-minimality of is a more basic result. It is essentially Gabrielov's theorem.
• More generally, is o-minimal. This is the expansion of 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 is piecewise continuous: the domain of can be written as a finite union of intervals, such that on each interval, is continuous. If , then one can also arrange that on each interval, is either constant, or strictly increasing, or strictly decreasing.
• Every definable subset of has finitely many definably connected components. In the presence of definable Skolem functions, each piece is definably path-connected.
• More precisely, every definable subset 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 is a definable function, then the domain of has a cell decomposition such that the restriction of to each cell is continuous.
• If , the topological closure of has dimension no bigger than , and the frontier has strictly lower dimension than .

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 for every . One also knows that if and are two definable functions , then and are asymptotically comparable. Limits always exist (possibly taking values ).

These results apply in particular to, e.g., the structure . In sharp contrast, the definable sets in 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… ::: ::: :::