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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!

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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
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@Nate: Why not post this as an answer? –  Jason DeVito Oct 14 '12 at 22:37

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up vote 5 down vote accepted

Here's the idea of a proof:

  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.

Another, similar approach would be to argue that that embedded submanifolds of lower dimension have Lebesgue measure zero.

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