A separable metric space is called fractal if its Hausdorff and topological dimensions are different.
The Hausdorff dimension is not invariant by homeomorphism (see this post).
Question: How to define a natural notion of Hausdorff homeomorphism which would imply that the Hausdorff dimension is invariant by Hausdorff homeomorphism?
Remark: By natural I mean not ad hoc, otherwise we can just define a Hausdorff homeomorphism as a homeomorphism between two spaces of same Hausdorff dimension, but it is not satisfying...