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.
Some relatively elementary examples:
By hard theorems of Alex Wilkie and other people, certain expansions of the ordered field 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 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.
Some analog of the Zilber trichotomy holds in the o-minimal setting.
Real algebraic geometry, Pila-Wilkie… ::: ::: :::