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.



Is there a concept of uniform Hausdorff dimension?
This is not an answer, just a comment:


I have never heard a name for the property you cite.


There is a significantly stronger property that is commonly used and does have a name: A metric space is called **Ahlfors $n$-regular** if there is a constant $K$ such that for each closed ball $B(x,r)$ in the space ($0<r\leq\text{diam}(X)$), 
$$ K^{-1} r^n \leq H^n(B(x,r)) \leq K r^n $$
where $H^n$ is $n$-dimensional Hausdorff measure.


In your example, $I$ and $D$ are of course Ahlfors $1$-regular and Ahlfors $2$-regular, respectively, while the union is not Ahlfors regular for any $n$.


One could consider any number of intermediate definitions between this concept and yours.


