We know that all connected subsets of $\mathbb{R}$( with the usual topology) has no empty interior. I would like to know if this fact remains true for a general 
 connected topological space with the Lebesgue covering dimension equal 1.



Topological spaces with Lebesgue covering dimension 1
If $C \subset [0,1]$ is the Cantor set, how about $([0,1] \times C) \cup (\{0\} \times [0,1])$ as a subset of $\mathbb{R}^2$? It's connected and one-dimensional, but $[0,1] \times \{0\}$ is nowhere dense.


