o-minimalityOMinimality2003-01-23 10:33:352003-02-11 09:35:54Definitiono-minimaldefinable setsemialgebraicreal algebraictame topology% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $M$ be an ordered structure. An interval in $M$ is any subset of $M$ that can be expressed in one of the following forms:
\begin{itemize} \item $\{x:a<x<b\}$ for some $a,b$ from $M$
\item $\{x:x>a\}$ for some $a$ from $M$
\item $\{x:x<a\}$ for some $a$ from $M$
\end{itemize}
Then we define $M$ to be {\em o-minimal} iff every definable subset of $M$ is a finite union of intervals and points. This is a property of the theory of $M$ i.e. if $M \equiv N$ and $M$ is o-minimal, then $N$ is o-minimal.
Note that $M$ being o-minimal is equivalent to every definable subset of $M$ being quantifier free definable in the language with just the ordering. Compare this with strong minimality.
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The model theory of o-minimal structures is well understood, for an excellent account see Lou van den Dries, Tame topology and o-minimal structures, CUP 1998.
In particular, although this condition is merely on definable subsets of $M$ it gives very good information about definable subsets of $M^{n}$ for $n \in \omega$.