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

I once learned that any countable dense subspace of any Euclidean space ℝ<sup>n</sup> 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 **J**<sup>n</sup> homeomorphic to **J** ?

Also, how far can this be pushed?

>**II**.   Let X be a dense totally disconnected subspace of ℝ<sup>n</sup> 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 ℝ<sup>n</sup> ?

>**IIa**.  Let X be any subspace of ℝ<sup>n</sup> as described in **II**., and let B denote any subspace of ℝ<sup>n</sup> homeomorphic to [the open unit ball in ℝ<sup>n</sup> 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 ℝ<sup>n</sup>) that are homeomorphic to **J** ?  What about **J**<sup>n</sup> ? (Perhaps the word *homogeneous* or *metric* needs to be included.)

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