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

I once learned that any countable dense subspace of any Euclidean space ℝⁿ is homeomorphic to the rationals ℚ.

Now I wonder if something similar is true for the irrationals J := ℝ - ℚ (with the subspace topology from ℝ).

Let c denote the cardinality of the continuum.

I. Is each cartesian power Jⁿ homeomorphic to J ?

Also, how far can this be pushed?

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

What about for such subspaces of fairly nice subspaces of ℝⁿ ?

IIa. Let X be any subspace of ℝⁿ as described in II., and let B denote any subspace of ℝⁿ homeomorphic to [the open unit ball in ℝⁿ union any subset of its boundary]. Then is X ∩ 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 ℝⁿ) that are homeomorphic to J ? What about Jⁿ ? (Perhaps the word homogeneous or metric needs to be included.)

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