I'm trying to see whether $\mathbb{Z}$ or the cantor set $C$ are submanifolds or $\mathbb{R}$.
Actually, I thought that $\mathbb{Z}$ was not a submanifold. As every subset of $\mathbb{Z}$ is countable while every open subset of $\mathbb{R}$ is uncountable, so there should be no diffeomorphism between them?
However, a quick research has yielded that this can't be true since $\mathbb{Z}$ is in fact a submanifold. But I didn't find proof for that.
Can anyone help me with that? Besides, is the cantor set a submanifold, too? By what argument?
[edit] Definition: $M$ is a $k$-dimensional submanifold of $\mathbb{R}^n$ iff for every $a \in M$ there are open subsets $U, V \subset \mathbb{R}^n$, such that $a \in U$, and a diffeomorphism $\phi: U \rightarrow V$ such that $\phi(M \cap U)=(\mathbb{R}^k \times \{0\}) \cap V.$
$\mathbb Z$ is a $0$-dimensional submanifold because every point has a neighborhood (obtained by intersecting with an open set in $\mathbb R$) that is diffeo to (an open set in) $\mathbb R^0$.
Can the same be said of the Cantor set?