Let $M$ be a metric space and let $U \subset M$ be open. Then the Hausdorff dimension of $U$ is defined in the usual way. If there is a single dimension number $d$ that is the Hausdorff dimension of every open set $U$ in $M$ I say the metric space $M$ has uniform Hausdorff dimension $d$.

What I want to know is whether some concept like this already exists, possible under a different name.

I want to use this as a property that 'nice' metric spaces have. A simple example: Let $I$ denote the unit interval and $D$ the unit ball in $\mathbb{R}^2$. Then both $I$ and $D$ have a uniform Hausdorff dimension but the disjoint union of $I$ and $D$ does not.