I was discussing orientability with a friend today. To me, orientation is a subtle concept I hardly understand. To get my perspective across, I was trying to come up with spaces which are intuitively orientable, but hard to orientate mathematically, and vice versa.
In this line of thought, I was wondering if there is a such general notion of orientability which allows the rational line $ℚ \subseteq ℝ$ or the rational circle $ℚ^2 ∩ S^1$ to be come orientable?
Let me get more precise (as there has been some discussion). I’m interested in a definition of orientability
(a) applicable to unusual, yet imaginable spaces like $ℚ$,
(b) fitting well with the intuition for these spaces, and
(c) generalising the usual notion for orientability for manifolds (if possible),
where I care less about (c) and more about (a) and (b). Alternatively, I would like to see an argument, based on intuition, for why such a notion isn’t sensible. I hope this isn’t too vague.