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

I once learned that any countable dense subspace of any Euclidean space ℝ^{n} 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 powerJ^{n}homeomorphic toJ?

Also, how far can this be pushed?

II. Let X be a dense totally disconnected subspace of ℝ^{n}such that every neighborhood of each point of X containscpoints. Is X homeomorphic toJ?

What about for such subspaces of fairly nice subspaces of ℝ^{n} ?

IIa. Let X be any subspace of ℝ^{n}as described inII., and let B denote any subspace of ℝ^{n}homeomorphic to [the open unit ball in ℝ^{n}union any subset of its boundary]. Then is X ∩ B homeomorphic toJ?

And what about greater generality ?

III. Is there a simple set of conditions that describe exactly all spaces (or subspaces of ℝ^{n}) that are homeomorphic toJ? What aboutJ^{n}? (Perhaps the wordhomogeneousormetricneeds to be included.)

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