We show that the class of locally finite varieties omitting type 1 has the following properties. This class (1) is definable by an idempotent, linear, strong Mal’cev condition in a language with one 4-ary function symbol. (2) is not definable by an idempotent, linear, strong Mal’cev condition in a language with only one function symbol of arity strictly less than 4. (3) is definable by an idemp...

Let R = (R,<,+, ·, . . . ) be a non-valuational weakly o-minimal real closed field, I a definable convex open subset of R and f : I → R a definable function. We prove that {x ∈ I : f ′(x) exists in R} is definable and f ′ is definable if f is differentiable.

Let U denote the Urysohn sphere and consider U as a metric structure in the empty continuous signature. We prove that every definable function U → U is either a projection function or else has relatively compact range. As a consequence, we prove that many functions natural to the study of the Urysohn sphere are not definable. We end with further topological information on the range of the defin...

Let R be an o-minimal expansion of a divisible ordered abelian group (R, <, +, 0, 1) with a distinguished positive element 1. Then the following dichotomy holds: Either there is a 0-definable binary operation · such that (R, <, +, ·, 0, 1) is an ordered real closed field; or, for every definable function f : R → R there exists a 0-definable λ ∈ {0} ∪ Aut(R, +) with limx→+∞[f(x) − λ(x)] ∈ R. Thi...

Given an o-minimal expansion M of a real closed field R which is not polynomially bounded. Let P∞ denote the definable indefinitely Peano differentiable functions. If we further assume that M admits P∞ cell decomposition, each definable closed set A ⊂ Rn is the zero-set of a P∞ function f : Rn → R. This implies P∞ approximation of definable continuous functions and gluing of P∞ functions define...

Let M be a structure in some language. Assume M has elimination of imaginaries. Let X be a definable set. Definable will mean “definable with parameters.” By a definable topology, we mean a definable family of subsets {By ⊂ X}y∈Y which form the basis for some topology on X. The fact that these form a basis for a topology amounts to the claim that if y1, y2 have By1 ∩By2 6= ∅, then for every x ∈...

An Euler characteristic χ is definable if for every definable function f : X → Y and every r ∈ R, the set {y ∈ Y : χ(f−1(y)) = r} is definable. If R = Z/nZ and M is a finite structure, there is a (unique) Euler characteristic χ : Def(M) → Z/nZ assigning every set its size mod n. This χ is always strong and ∅definable. If M is an ultraproduct of finite structures, then there is a canonical stron...

In 1934, Whitney gave a necessary and sufficient condition on a jet of order m on a closed subset E of R to be the jet of order m of a C-function; jets satisfying this condition are known as C-Whitney fields. Later, Paw lucki and Kurdyka proved that subanalytic C-Whitney fields are jets of order m of sybanalytic C-functions. Here, we work in an o-minimal expansion of a real closed field and pro...