Are there examples of homeomorphic compact metric spaces of different Hausdorff dimension?
If yes, are there sufficient conditions on the spaces which would imply the equality of Hausdorff dimensions?
In my case the spaces are the so called multiple conic singularities (MCS-) spaces. They are defined inductively: MCS-spaces of dimension -1 are empty. Any point of an $n$-dimensional MCS-space has an open neighborhood homeomorphic to an open cone over its boundary which is a compact $(n-1)$-dimensional MCS-space. (Such examples come from the theory of Alexandrov spaces.)