ON THE ASSOUAD DIMENSION AND CONVERGENCE OF METRIC SPACES

We introduce the notion of pseudo-cones of metric spaces as a generalization of both of the tangent cones and the asymptotic cones. We prove that the Assouad dimension of a metric space is bounded from below by that of any pseudo-cone of it. We exhibit an example containing all compact metric spaces as pseudo-cones, and examples containing all proper length spaces as tangent cones or asymptotic cones.