Arbitrarily High Hausdorff Dimensions of Continua

Abstract

It is well-known that Hausdorff dimension is not a topological invariant; that is, that two homeomorphic continua can have different Hausdorff dimension, although their topological dimension will be equal. We show that it is possible to take any continuum embeddable in $\mathbb{R}^n$ and embed it in such a way that its Hausdorff dimension is $n$. In doing so, we can obtain an arbitrarily high Hausdorff dimension for any nondegenerate continuum. As an example, we will give different embeddings of an arc whose Hausdorff dimension is any real number between $1$ and $\infty$, including an arc of infinite Hausdorff dimension.