(This question actually arose in some research on number theory.)
I once learned that any countable dense subspace of any Euclidean space ℝn$\mathbb R^n$ is homeomorphic to the rationals ℚ$\mathbb Q$.
Now I wonder if something similar is true for the irrationals J := ℝ - ℚ$J \mathrel{:=} \mathbb R \setminus \mathbb Q$ (with the subspace topology from ℝ$\mathbb R$).
Let c$\mathfrak c$ denote the cardinality of the continuum.
I. Is each cartesian power Jn$J^n$ homeomorphic to J $J$?
Also, how far can this be pushed?
II. Let X$X$ be a dense totally disconnected subspace of ℝn$\mathfrak R$ such that every neighborhood of each point of X$X$ contains c$\mathfrak c$ points. Is X$X$ homeomorphic to J $J$?
What about for such subspaces of fairly nice subspaces of ℝn $\mathbb R^n$?
IIa. Let X$X$ be any subspace of ℝn$\mathbb R^n$ as described in II., and let B$B$ denote any subspace of ℝn$\mathbb R^n$ homeomorphic to [the open unit ball in ℝn union$\mathbb R^n$ $\cup$ any subset of its boundary]. Then is X ∩ B$X \cap B$ homeomorphic to J $J$?
And what about greater generality ?
III. Is there a simple set of conditions that describe exactly all spaces (or subspaces of ℝn$\mathbb R^n$) that are homeomorphic to J $J$? What about Jn $J^n$? (Perhaps the word homogeneous or metric needs to be included.)
(I found nothing relevant via Google, in MathSciNet, or here on MathOverflow.)
