Let $n\in \mathbb{N}$ and $A\subseteq \mathbb{R}^n$ be a subset such that its Hausdorff dimension is strictly less than $n-1$. Is it then true that $\mathbb{R}^n\setminus A$ is (path-)connected? Intuitively if we consider $\mathbb{R}^2$, then we need at least a line (i.e. something $1$-dimensional) to disconnect the space. Similarly in $3$-dimensions one would require something $2$-dimensional and so on.
In my particular case I have a subset which is a countable union of smoothly embedded $1$-manifolds in $\mathbb{R}^3$ and my intuition tells me the complement must be path-connected, but I was not able to put my intuition into a rigorous proof.
The problem seems rather intuitive, so I guess there will be some results in the literature already? If anyone has a hint I would appreciate it.
Kind Regards Dennis
