In this paper, we propose a combinatorial approach to study Calabi–Yau threefolds constructed as a resolution of singularities of a double covering of [Formula: see text] branched along an arrangement of eight planes. We use this description to give a complete classification of arrangements of eight planes in [Formula: see text] defining Calabi–Yau threefolds modulo projective transformation with [Formula: see text] and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).

Abstract. The aim of this note is to give yet another proof of the following theorem: given an arbitrary o-minimal structure on the ordered field of real numbers R and any definable family A of definable nonempty compact subsets of R n , then the closure of A in the sense of the Hausdorff metric (or, equivalently, in the Vietoris topology) is a definable family. In particular, any limit in the sense of the Hausdorff metric of a convergent sequence of subsets of a definable family is definable in the same ominimal structure.

We give a geometric and elementary proof of the uniform$\mathscr{C}^{p}$-parametrization theorem of Yomdin and Gromov in arbitrary o-minimal structures.

In this note we prove a semialgebraic counterpart of Whitney's extension theorem.