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A structure with a linear order is o-minimal if every definable set in one variable is a finite union of intervals and points. O-minimality is an extremely strong tameness condition: definable sets have controlled geometric structure, cell decomposition holds, and the theory is decidable. O-minimal structures include (ℝ, <, +, ·) and expansions with analytic functions.
From your work on quantifier elimination, you know that certain structures can eliminate quantifiers — reducing any definable set to one described by a quantifier-free formula. In the real ordered field (ℝ, <, +, ·), quantifier elimination gives Tarski's theorem: every semi-algebraic set (defined by polynomial inequalities) can be described without quantifiers. O-minimality abstracts and strengthens this idea. A structure (M, <, ...) is o-minimal (order-minimal) if every subset of M that is definable (using any formula, possibly with parameters) is a finite union of open intervals and points. This is the tamest possible behavior for definable subsets of the line: no Cantor sets, no complicated open sets, no fractals — just finitely many pieces, each of which is an interval or an isolated point.
The condition is imposed only on one-variable definable sets, but it propagates to all dimensions through the cell decomposition theorem: in an o-minimal structure, every definable subset of Mⁿ can be partitioned into finitely many cells, which are smooth (differentiable) sets built inductively from one-dimensional pieces. A cell in M¹ is just a point or open interval. A cell in M² is either a "graph" (a definable continuous function's graph) or a "band" (the region between two such graphs over a cell in M¹). This inductive geometric structure means that o-minimal sets are stratified and well-behaved in ways that general definable sets in, say, number theory are catastrophically not. In particular, o-minimal sets have finitely many connected components, and their Euler characteristics and Betti numbers are uniformly bounded in terms of the formula defining them.
The canonical o-minimal structure is (ℝ, <, +, ·) — the real closed field. Here o-minimality follows from quantifier elimination for real closed fields (Tarski-Seidenberg): every definable set is semi-algebraic, and a semi-algebraic subset of ℝ is always a finite union of intervals and points. More exotic o-minimal structures include (ℝ, <, +, ·, exp) where exp is the real exponential function — this is the content of Wilkie's theorem (1996), a deep result showing that the expansion of the reals by exp remains o-minimal, even though exp is transcendental. This allows definable sets to include graphs of exponential, logarithmic, and power functions while still maintaining the cell decomposition and tameness properties.
O-minimality matters for tame geometry: a program, largely due to Grothendieck, Pillay, and Steinhorn, of replacing ad hoc tameness conditions in analysis and topology with the single structural assumption of o-minimality. Classical results like the Morse lemma, triangulation of manifolds, and finiteness of topological types all generalize cleanly to o-minimal structures. From the model-theoretic perspective, o-minimality is a form of stability-adjacent tameness — it is not a stability condition (o-minimal structures are generally unstable because of the linear order) but it achieves comparable structural control through geometric rather than combinatorial means. If stability theory is about algebraic tameness, o-minimality is about geometric tameness.