(4.1.1) The case where $X$ is a family of Calabi-Yau varieties over $C$ is of particular interest; the connections with homological mirror symmetry were the main motivation for Kontsevich and Soibelman to define the skeleton in [KS06]. If $\omega$ is a volume form on $X_{K}$ (i.e., a nowhere vanishing differential form of maximal degree) then $\mathrm{Sk}(X_{K})=\mathrm{Sk}(X_{K},\omega)$ by [MN13, 4.6.4]. We will now prove that the underlying topological space of the essential skeleton $\mathrm{Sk}(X_{K})$ is a pseudo-manifold with boundary; this result is implicitly contained in [KK10, Ko11].