Dugundjiʹs canonical covers, asymptotic and covering dimension

Given a nowheredense closed subset X of a metrizable compact space X ˜ , we characterize the dimension of X in terms of the multiplicity of the canonical covers of the complementary of X, especially in some particular cases, like when X ˜ is the Hilbert cube or the finite dimensional cube and X, a Z-set of X ˜ . In this process, we solve some related questions in the literature.