Is $\mathbb{P}^n(\mathbb{Q})$ a (smooth) manifold? I think it is a real differentiable manifold, but what is its dimension? 0? Must the topology on $\mathbb{Q}$ as a topological manifold, be the discrete topology? What is the (real) dimension of $\mathbb{Q}$? Of $\mathbb{P}^n(\mathbb{Q})$? Or, must we only speak of $\mathbb{Q}$-dimension? Do features of $\mathbb{Q}\subset\mathbb{R}$ carry over to $\mathbb{P}^n(\mathbb{Q})\subset\mathbb{R}$, such as:
- Does $\mathbb{P}^n(\mathbb{Q})$ have `measure zero' in $\mathbb{P}^n(\mathbb{R})$ (is there an appropriate measure here?)? (Does there exist $n\in\mathbb{N}$ such that $\mathbb{P}^n(Q)$ has positive volume?)
- Is $\mathbb{P}^n(\mathbb{Q})$ still dense in $\mathbb{P}^n(\mathbb{R})$?
- Does $T\mathbb{Q}^n$ have $\mathbb{Q}$-dimension 2n, and real dimension 0?
- Is $\mathbb{P}^n(\mathbb{Q})$ orientable for $n$ odd?
- Is $\mathbb{P}^n(\mathbb{Q})$ homeomorphic to $\mathbb{P}^1(\mathbb{Q})$?
(This is a stretch, if the others haven't been): Can viewing $\mathbb{P}^n(\mathbb{Q})$ as a manifold answer any (admittedly, elementary) questions in number theory?
Any books or references on this subject would be much appreciated, if they exist. Thanks!