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> **Question 1.** Given a topological space $X$ and two metrics $a$ and $b$ on it, compatible with the topology, what conditions should I impose on them so that box-counting (or other, for example Hausdorff) dimensions of $(X,a)$ and $(X,b)$ are equal?
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> **Question 2.** Is there a notion of a dimension for topological (as opposed to metric) spaces which can assume non-integral values?
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My motivation
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Let $G$ be a finitely generated group and let $p$ be a prime number. Consider the compact topological space $X:=\prod\_G \mathbb Z/p$, infinite product of copies of the cyclic group of order $p$, indexed by elements of $G$. Let $T$ be an element of the integral group ring of $G$. Note that $T$ gives a map $X\to X$ in a natural way (in this context $T$ is sometimes called *cellular automaton*). Define $Y$ to be the subset of those points $x$ of $X$ such that $T(x)=0$.


I want to measure "how big" $Y$ is. 


If we choose a generating set for $G$ then we get a metric $d$ on $G$, and we also get a metric on $X$: two sequences $x\_i$ and $y\_i$ of $X$ are $p^{-|B(1,k)|}$ apart if they agree on the ball $B(1,k)$ of radius $k$ around the neutral element $1$ of $G$, but they don't agree on any larger ball. It's straightforward to see that box counting dimension of $X$ is $1$.


Unfortunately the metric we just defined depends on the generators chosen for $G$, so I'd like to know if I have a chance to get any kind of "intrinsic size of $Y$" this way. 



How indepenedent of a chosen metric is the box-counting dimension? Is there a non-integral dimension which is defined for topological spaces?
**Q2**. If you want that dimension for topological spaces to agree with Hausdorff dimension (for example) in case of metric space, then NO. For any $0 \le s \le \infty$, there is a metric on the Cantor set so that the Hausdorff dimension is $s$.


**Another topological result**. Let $X$ be a separable metrizable space. The infimum of the Hausdorff dimensions of all metric spaces homeomorphic to $X$ is the topological dimension of $X$. (Integer valued.)


**for the My Motivation comment**. See the notion "topological entropy".


