Author Archives: scanlon

Every o-minimal structure admits an Euler characteristic with values in $\mathbb{Z}$.  Our computation showing that $K_0(\mathfrak{R})$ is a quotient of $\mathbb{Z}$  when $\mathfrak{R}$ is an o-minimal expansion of an ordered field may be reversed to define the o-minimal Euler characteristic.

If $\mathfrak{R}$ is an o-minimal expansion of an ordered field, then $K_0(\mathfrak{R})$ is a quotient of $\mathbb{Z}$.  In a later post, we will see that, in fact, $K_0(\mathfrak{R})$ is exactly $\mathbb{Z}$.

We have seen how to assign a natural number valued dimension to definable sets in o-minimal structures.  There is a second integer valued invariant, which we will call the Euler characteristic.   In o-minimal expansions of fields, these two invariants characterize … Continue reading →

A vector space $V$ with the closure operation defined by taking a subset $A \subseteq V$ to the linear span of $A$ is a quintessential example of a pregeometry.  To any vector space $V$ we may define the dimension of … Continue reading →

For the sake of building a dimension theory for definable sets in o-minimal structures, we will  follow a more model theoretic approach, due to Pillay, to dimensions than what appears in van den Dries’ book.  That is, we will show … Continue reading →

In our proof of the Cell Decomposition Theorem, we took as known the Uniform Finiteness Theorem.  In this post, we fill that gap. Uniform Finiteness Theorem:   Let $Y \subseteq R^{n+1}$ be a definable set for which all fibers $Y_a := … Continue reading →

The Cell Decomposition Theorem might be called the “Fundamental Theorem of O-minimality”.   With this theorem we show that from the hypothesis that the definable sets in one variable are particularly simply, i.e. those which admit a quantifier-free definition in the … Continue reading →

A basic principle in tame geometry is that there are no pathological definable functions in o-minimal structures.   One precise sense in which this principle is true is given by the Monotonicity Theorem. Monotonicity Theorem :  Let $f: R \to R$ … Continue reading →