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 $\{x\in M:x\leq a\}$ and $\{x\in M:x\geq a\}$ . 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… ::: ::: :::