## FANDOM

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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 Edit

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 Edit

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 Edit

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

## Applications Edit

Real algebraic geometry, Pila-Wilkie…