TITLE: Are countable dense subspaces of $\mathbb{R}^n$ homeomorphic to ${\mathbb Q}^n$?
QUESTION [12 upvotes]: Let $n\geq 1$ be an integer and suppose $S\subseteq {\mathbb R}^n$ is countable and dense. Do we have $S \cong {\mathbb Q}^n$ where both sets carry the topology inherited from the Euclidean topology on ${\mathbb R}^n$?

REPLY [9 votes]: You can learn a bit more about countable dense subsets of separable metric spaces by searching for "countable dense homogeneous space".
A separable metric space $X$ is Countable Dense Homogeneous (CDH)
if given any two countable dense subsets $D$ and $E$ of $X$
there is a homeomorphism $f : X \rightarrow X$ such that
$f(D) = E$.
The concept was introduced by R. Bennett in
Countable dense homogeneous spaces
Fund. Math., 74 (1972), pp. 189-194
Theorem 3 in the paper implies that locally euclidean spaces are CDH.
So not only any countable dense subset of ${\bf R}^n$ is homeomorphic to 
${\bf Q}^n$, but the homeomorphism can be taken as the restriction
of a global homeomorphism of ${\bf R}^n$.