I would like to know if it is possible to cover $\mathbb{R}^n$ by countably many immersed submanifold of dimension less than $n$. A similar version is whether it is possible to cover $\mathbb{C}^n$ by countably many analytic subsets of lower dimension.
The motivation is that an exercise I am working involves in proving a statement being true for a generic lattice, which seems to invoke statements of the sort above, but I am not sure how I can prove them. Thanks!