Please, explain:
the Cantor set (a zero-dimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is $\ln2 / \ln3$, which is approximately 0.63 The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of $\ln3 / \ln2$, which is approximately 1.58.
Source Wikipedia.
I understand this in a way that topological dimension is a measure of how to discriminate objects, more here, $\dim_\text{topo}(\emptyset) = -1$, $\dim_\text{topo}(\cdot)=0$ because you need nothing to discriminate point, $\dim_\text{topo}(|)=1$ because you need a point to discriminate a line. Similarly for:
- $\dim_\text{topo}(\#)= 1$ because you need four points to discriminate it and the supremum of the local dimesion is 1.
- $\dim_\text{topo}(\text{keyboard}) = 3$ because I need a plane to discriminate it.
But what about them, how can I use similar logic as above to discriminate them?
- $\dim_\text{topo}(\text{Cantor sets}) = ?$
- $\dim_\text{Hausdorff}(\text{Cantor sets}) =?$