Journal of the
Korean Mathematical Society JKMS

We prove that a non-empty separable metrizable space \par \noindent $X$ admits a totally bounded metric for which the metric dimension of~$X$ in Assouad's sense equals the topological dimension of~$X$, which leads to a characterization for the latter. We also give a characterization based on this Assouad dimension for the demension (embedding dimension) of a compact set in a Euclidean space. We discuss Assouad dimension and these results in connection with porous sets and measures with the doubling property. The elementary properties of Assouad dimension are proved in an appendix.

Keywords: topological dimension, metric dimension, demension, fractal, porous, homogeneous measure, doubling property, Mobius

MSC numbers: Primary 54F45; Secondary 28A75