?. , Recall that for every x ? X the local dimension dim(X, x) is the minimum of dim U ? X as U ranges over the definable neighbourhood of x. Let W k (X) denote the set of x ? X such that there is a definable neighbourhood B of x and a coordinate projection ? : K m ? K k which induces by restriction a homeomorphism between B ? X and an open subset of K k. We say that K is a tame topological structure if it satisfies the following properties, for every definable sets X, Y ? K m and every definable function f : X ? K n

, Dim2: dim X ? Y = max(dim X, dim Y )

, Dim3: dim(X) = dim(X) and if X = ?, then dim(X \ X) < dim(X)

, Example 11.4 Ever o-minimal, C-minimal or P-minimal expansion of a field K is tame (see, vol.16

, More generally, every dp-minimal expansion of a field K which is not strongly minimal is tame (see [14]). Following [8] we may also consider the models of visceral theories having finite definable choice and no space-filling function: all of them are tame. This applies in particular, with the interval topology, to every divisible ordered Abelian group whose theory is weakly o-minimal. Note that by (Dim1), dim f (X) = dim X if f is bijective

, X has pure dimensional if and only if X = ? d (X) with d = dim X. The sets ? k (X) form a partition of X. For every k ? 0, l?k ? l (X) is closed in X (for every k)

, Proposition 11.5 With the above notation and assumptions, W k (X) is a dense subset of ? k (X)

, If non-empty, they have dimension k. In particular, X has pure dimension k if and only if W k (X) is non-empty and dense in X

, We turn now to density. Pick x ? ? k (X) and a neighbourhood U of x in X. By shrinking U if necessary we may assume that dim U = k. From (Dim4) we know that W k (U ) = ?. On the other hand, W k (U ) ? W k (X) because U is open in X. Consequently W k (U ) ? B ? W k (X) and so B ? W k (X) = ?. This proves density. By (Dim3) we have dim ? k (A) = dim W k (A), so it only remains to check that W k (X) has dimension k, provided it is not empty. Clearly dim W k (X) ? k. If dim W k (X) = l > k then by (Dim4) W l (W k (X)) is non-empty. But W k (X) is open in X, hence W l (W k (X)) is contained in W l (X). So W l (W k (X)) is contained both in W l (X) and in W k (X), a contradiction since W l (X) and W k (X) are disjoint (they are contained in ? k (X) and ? l (X) respectively). The last point follows, since X has pure dimension k if and only if X = ? k (X) = ?, Proof: If x ? W k (X), there is a definable neighborhood U of x in X, a coordinate projection ? : K m ? K k and an open subset V of K k such that ? induces by restriction a homeomorphism between U and V. In particular dim U = k by (dim1), hence dim W k (X) ? k

K. Let and .. , ) be a tame topological structure, and X ? K m be a definable set. 1. For every A ? L, dim L def (X) A = dim A

, L def (X) is a d-scaled lattice

, The first item ensures that A ? L is k-pure in L def (A) if and only if it is so in the geometric sense

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