I consider a bounded open set $A$ in ${\mathbb R}^d$. Is the Hausdorff dimension of the boundary of $A$ at least $d-1$ ? I thought I would have found a result on this problem in any textbook about Hausdorff dimension but I failed. As you may guess I have never work with Hausdorff dimension.



Hausdorff dimension of the boundary of an open set in the Euclidean space - lower bound
If a compact set $C$ separates a connected space $X$ of dimension $d$ (with some homogeneity properties, for example, an euclidean space) in two connected components, then it must have topological dimension at least $d-1$ (see for example the book by Hurewicz and Wallman on topological dimension, Theorem IV.4). In general, [Hausdorff dimension](http://en.wikipedia.org/wiki/Hausdorff_dimension) exceeds (or is equal to) topological dimension (see chapter VII.2 of the book by Hurewicz-Wallman mentioned above).


