Suppose $X\subseteq \mathbb R^2$ is nowhere compact ($X$ has no compact neighborhood) and non-empty.

Can $X$ be *densely* embedded into the plane?

In other words, is there a dense set $X'\subseteq \mathbb R ^2$ such that $X'\simeq X$?

I believe the answer is yes if $X$ is zero-dimensional. Is it still true for totally disconnected $X$?

Also I am primarily interested in Polish spaces.