It's known that a zero-dimensional, compact metric space $X$ embeds in the Cantor Set: Take a countable, clopen basis $(U_j)$ indexed by $\mathbb{N}$ and construct a map $f: X \rightarrow \lbrace 0, 1 \rbrace^\mathbb{N}$ such that the $j$th coordinate of $f$ be the characteristic function of $U_j$.
If $X$ is instead presumed to be embedded in some space $Y$, then you can't assume there is a Cantor Set in $Y$ containing $X$; for example, take $X = Y = $ a point.
But in the plane, unless I am having some major malfunction there is always a way. In fact, you can even construct it explicitly, but I'm wondering if there's a slick, formal proof of this fact without getting into any messy constructions.
Thanks!