Proof If Cexp is model complete, then every definable set is a projection of a closed set. Since C is locally compact, every definable set is Fσ. The same is true for the complement, so every definable set is also Gδ. But, since Z is definable, Q is definable and a standard corollary of the Baire Category Theorem tells us that Q is not Gδ. Still, there are several interesting open questions abo...

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an a...

Abstract We obtain some new results on the topology of unary definable sets in expansions densely ordered Abelian groups burden 2. In special case which structure has dp‐rank 2, we show that existence an infinite discrete set precludes definability a is dense and codense interval, or topologically like Cantor middle‐third (Theorem 2.9). If it 2 both D dense‐codense X are definable, then transla...

The open core of a structure R := (R, <, . . .) is defined to be the reduct (in the sense of definability) of R generated by all of its definable open sets. If the open core of R is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of R is finite or uncountable, or if R defines addition and multiplication ...

In this work we answer a question stated by Loi and Zaharia concerning trivialization of definable functions off the bifurcation set: we prove that definable functions are trivial off the bifurcation set, and the trivialization can be chosen definable.

Let G be a definably compact definable C group and 1 ≤ r < ∞. Let X be a definable CG submanifold of a representation of G and Y a definable C submanifold of R. We prove that every G invariant surjective submersive definable C map f : X → Y is piecewise definably CG trivial.

The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. In this paper we present a framework of graph polynomials based on counting functions of generalized colorin...

This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure 〈H(κ+),∈〉 by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the i-function and construct a c...