Model Theoretic Definition
O-minimal structures originated in model theory and so have a simpler — but equivalent — definition using the language of model theory. Specifically if L is a language including a binary relation <, and (M,<,...) is an L-structure where < is interpreted to satisfy the axioms of a dense linear order, then (M,<,...) is called an o-minimal structure if for any definable set X ⊆ M there are finitely many intervals I1,...,Ir with endpoints in M ∪ {±∞} and a finite set X0 such that
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