# How to define a notion of Hausdorff homeomorphism?

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...

• A comment by @DanielFisher here suggests bi-Lipschitz homeomorphisms are what you need. Are you familiar with this class of maps? – Eric Towers May 21 at 15:42
• @EricTowers: not that much but I will look for it. I just found this reference also stating this result (in Introduction) but without proof or reference. I wonder whether there is a reference with a proof or if it is just an exercise. Anyway it is the natural notion I needed, thanks! – Sebastien Palcoux May 21 at 16:10
• This is indeed just an exercise in following the definition of the Hausdorff measure. – Moishe Kohan May 24 at 10:51