Let $\mu^n$ be the Menger compactum for a natural number $n$. It is known that $\mu^0=C$, where $C$ denotes the Cantor set. Let $\ast$ denote the join of topological spaces.
Question: Is it true that $\mu^{n+1}\subset\mu^n\ast C$ as topological spaces?
Take $n=0$. Then at least the construction of $\mu^0=C$ and $\mu^1$ can be visualized. However, it is still unclear for me if $\mu^1\subset C\ast C$.
Let me just write that the answer for my question is "no". Take the case $n=0$. Every open set in $\mu^1$ contains a circle and every point in $C\ast C$ has a neighborhood that is not contaning a circle.