# Is the Hausdorff dimension invariant under homeomorphisms?

Recently I was thinking extensively about dimensions of a spaces and ascertained that algebraic or topological definitions of dimensions could hardly be not an integer.

So I'm curious whether the Hausdorff or Minkowski dimensions invariant under homeomorphisms, not just isometries for if not, it is not a measurement of an intrinsic property of the space under consideration.

Does anyone has a good idea on this?

• A middle-$\alpha$ Cantor set has Hausdorff dimension $\frac{\log 2}{\lvert \log \alpha\rvert}$, and one has homeomorphisms $\mathbb{R}\to \mathbb{R}$ that map it to the standard middle-third Cantor set. So Hausdorff dimension isn't invariant under homeomorphisms. [Not surprisingly. We need bi-Lipschitz homeomorphisms for invariance of Hausdorff dimension.] – Daniel Fischer Dec 16 '16 at 14:49
• @DanielFischer Thanks. Although I didn't plunged into a text yet, but it helped me to get a reasonable intuition. – HyJu Dec 16 '16 at 15:51