Every totally disconnected separable metric space of dimension $n$ homeomorphically embeds into $C\times \mathbb R ^n$.

Is something like this known? $X$ is totally disconnected means that every point in $X$ is equal to the intersection of all clopen sets containing the point. $C$ is the Cantor set.

It is known that there are totally disconnected spaces of arbitrary dimension. But what about just $n=1$? How might we prove $X$ embeds into $C\times [0,1]$?