(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.)