Let $C$ be the middle-thirds Cantor set. Obviously $C\times [0,1]$ embeds into the plane. But $C\times D$ does not, $D$ being a closed disc in the plane.

Are there any general results which can be applied to sets like this (Cantor set times a plane set) to see if they do, or do not, embed into the plane?

Is there a 1-dimensional continuum $X\subseteq \mathbb R ^2$ such that $C\times X$ does *not* embed into the plane? How about if $\mathbb R ^2\setminus X$ is path-connected?