A structure {\displaystyle (M,<,\ldots )} is said to be o-minimal if every subset {\displaystyle X\subset M^{1}} definable with parameters from {\displaystyle M} can be written as a finite union of points and intervals, i.e., as a boolean combination of sets of the form {\displaystyle \{x\in M:x\leq a\}} and {\displaystyle \{x\in M:x\geq a\}}. Note that this is an assertion about subsets of {\displaystyle 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 {\displaystyle T} with a predicate {\displaystyle <} is said to be o-minimal if every model of {\displaystyle 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 {\displaystyle (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.


Some relatively elementary examples:

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


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:

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 {\displaystyle C^{k}} for every {\displaystyle k<\infty }. One also knows that if {\displaystyle f} and {\displaystyle g} are two definable functions {\displaystyle \mathbb {R} \to \mathbb {R} }, then {\displaystyle f} and {\displaystyle g} are asymptotically comparable. Limits always exist (possibly taking values {\displaystyle \pm \infty }).

These results apply in particular to, e.g., the structure {\displaystyle (\mathbb {R} ,+,\cdot ,\exp )}. In sharp contrast, the definable sets in {\displaystyle (\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.


Real algebraic geometry, Pila-Wilkie… ::: ::: :::