DEFINABLY CONNECTED NONCONNECTED SETS ANTONGIULIO FORNASIERO Abstract. We give an example of a structure R on the real line, and a mani- fold M definable in R, such that M is definably connected but is not connected. 1. Introduction Let R be an expansion of the real field ¯ R := 〈R, +, ·,<〉. Let M be a definable subset of R. We remind that M is called definably connected if there is no clopen definable subset Y of M such that ∅ = Y = M . Main Theorem. There exists a structure R expanding ¯ R and set M definable in R, such that: (1) M is a 1-dimensional embedded C 1 submanifold of R 3 , (2) M has 2 connected components, (3) M is definably connected. We know that such R cannot be o-minimal, because in an o-minimal expansion of R every definable and definably connected set is arc-connected (and hence con- nected). However, we can find R as above which is also d-minimal (see [Mil05] for the definition and main properties of d-minimal structures). The main ingredient is the following result, which follows easily from the proof of [MT06, Theorem 1]; we will give some details of the proof in §3. Lemma 1.1. There exists a sequence P = 〈c n : n ∈ N〉 of real numbers, such that (1) P is strictly increasing and unbounded; (2) the set Q := {c n : n even} (as a set ) is not definable in R, the expansion of ¯ R with a new predicate for P (where P is also regarded as a set ). Moreover, we can find P as above such that R is also d-minimal. We will show that R as in the above lemma satisfies the conclusion of the Main Theorem. 1.1. Application to Pfaffian functions. Let R and M be as in the proof of the Main Theorem. In [Fra06], S. Fratarcangeli introduced the relative Pfaffian closure of ¯ R inside R. Consider the following 1-form on R 3 ω(x, y, z) := dz. Let L be the xy-coordinate plane. Notice that L is a Rolle Leaf with data 〈R 2 ,ω〉, according to the definition in [Spe99]. Let X 0 be the translate of M along the z-axis, such that the endpoints of M are on L, X 1 be its mirror image along the xy-plane, and X := X 0 ∪ X 1 . Then, X is a 1-dimensional C 1 manifold which is definable in R and definably connected, which intersects L in 2 points, but which is never orthogonal to ω. Thus, L is not a R-Rolle Leaf, according to [Fra06, Definition 5.2]: therefore, it is not always the case that a Rolle Leaf (à la Speissegger) definable in R and with data definable in ¯ R is a R-Rolle Leaf à la Fratarcangeli. Date : 18 Mar 2011. 2010 Mathematics Subject Classification. Primary 03Cxx; Secondary 12J15, 03C64. Key words and phrases. Connected, definably connected. Thanks to Immanuel Halupczok and Philipp Hieronymi for their help in writing this article. 1