I don't know how to finish this one. Here's what I've done so far:

We're trying to find a contradiction.

Suppose that $\mathbb{R^n}=C_1\cup C_2 \cup \dots \cup C_n$, with $C_i$ closed subsets (with empty interior) of $\mathbb{R^n}$.

It follows that $C_1^c \cap C_2^c \cap \dots \cap C_n^c = \emptyset$.

I've got to use somehow that $Int(C_i)=\emptyset$ but I can't figure out how.

EDIT I've been only defined the closed-open set concept, neighborhood, closure and interior.

I'd appreciate some hint to finish this one.

Thanks for your time.

  • 1
    $\begingroup$ This is another version of the completeness of $\mathbb{R^n}$. See Baire's Catagory Theorem. $\endgroup$ – Arjun Banerjee Oct 25 '18 at 11:33
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    $\begingroup$ @ArjunBanerjee: That would be if the union was countable rather than finite. For a finite union it holds in arbitrary spaces. $\endgroup$ – Henning Makholm Oct 25 '18 at 12:01

Can you prove that the union of two closed sets with empty interior is itself a closed set with empty interior?

If so, then use induction to extend this to "finitely many".

(I find it slightly easier to think of the complements instead and prove that the intersection of two dense open sets is dense and open. Your mileage may vary, though).

  • $\begingroup$ Can I think of a closed set with an empty interior as a set with only its boundary points, so that any sequence defined on the boundary of the set converges to a boundary point? $\endgroup$ – Arjun Banerjee Oct 25 '18 at 11:46
  • $\begingroup$ @Henning I'm trying it but it's not trivial with only the definitions. Could you add a sketch? $\endgroup$ – Relure Oct 25 '18 at 11:52
  • $\begingroup$ @Relure: One further hint then: Prove that the intersection of a dense open set with a dense set is dense. $\endgroup$ – Henning Makholm Oct 25 '18 at 12:00

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