# Covering $\mathbb{R}^n$ by countably many lower dimensional pieces?

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!

-
Idea: (1) embedded submanifolds of lower dimension are nowhere dense; (2) an immersed submanifold is a countable union of embedded submanifolds; (3) the Baire category theorem. – Nate Eldredge Oct 14 '12 at 19:13
Ah, Baire's theorem of course! Thanks a lot. – user27126 Oct 14 '12 at 19:18
@Nate: Why not post this as an answer? – Jason DeVito Oct 14 '12 at 22:37