(This question actually arose in some research on number theory.)

I once learned that any countable dense subspace of any Euclidean space $\mathbb R^n$ is homeomorphic to the rationals $\mathbb Q$.

Now I wonder if something similar is true for the irrationals $J \mathrel{:=} \mathbb R \setminus \mathbb Q$ (with the subspace topology from $\mathbb R$).

Let $\mathfrak c$ denote the cardinality of the continuum.

I. Is each cartesian power $J^n$ homeomorphic to $J$?

Also, how far can this be pushed?

II. Let $X$ be a dense totally disconnected subspace of $\mathfrak R$ such that every neighborhood of each point of $X$ contains $\mathfrak c$ points. Is $X$ homeomorphic to $J$?

What about for such subspaces of fairly nice subspaces of $\mathbb R^n$?

IIa. Let $X$ be any subspace of $\mathbb R^n$ as described inII., and let $B$ denote any subspace of $\mathbb R^n$ homeomorphic to [the open unit ball in $\mathbb R^n$ $\cup$ any subset of its boundary]. Then is $X \cap B$ homeomorphic to $J$?

And what about greater generality?

III. Is there a simple set of conditions that describe exactly all spaces (or subspaces of $\mathbb R^n$) that are homeomorphic to $J$? What about $J^n$? (Perhaps the wordhomogeneousormetricneeds to be included.)

(I found nothing relevant via Google, in MathSciNet, or here on MathOverflow.)