I have been looking for certain properties of the Hausdorff dimension lately, and noticed that all the examples I know of in $\mathbb{R}$ are only totally disconnected spaces. So I was wondering whether any subset $A\subset \mathbb{R}$ satisfying $\dim(A)<1$ must be disconnected or even totally disconnected?
Are there counter-examples to these 'conjectures'? Or perhaps some known results on this subject which are relevant?
If $A\subset \mathbb R$ and $\dim A <1$, then $A$ has to be totally disconnected: if not, then one of the connected component of $A$ is not a one point set. Then that connected component contains an interval, which contradicts to the assumptions that $\dim A <1$.
Note that there are totally disconnected subset $B$ of $\mathbb R$ with $\dim B = 1$: for example $B = \mathbb R\setminus \mathbb Q$.