Q:

Nowhere compact subsets of the plane

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.

A:

According to the definition here a topological space is nowhere compact if every compact subset has empty interior. Assuming that's that you mean by "nowhere compact", the following set $X$ seems to be a counterexample for your first question, without "totally disconnected":
$$X=\{(x,y)\in\mathbb R\times\mathbb R:x^2+y^2\le1\}\setminus\{(x,y)\in\mathbb Q\times\mathbb Q:x^2+y^2\lt1\}$$
Let $S=\{(x,y)\in\mathbb R\times\mathbb R:x^2+y^2=1\}\subset X$. If $X$ is embedded in the plane, then the image of $S$ is a simple closed curve. Since $X\setminus S$ is connected, its image is either entirely inside or entirely outside that simple closed curve. In either case, the image of $S$ is not dense in the plane.

