For the last three problems, we consider the following construction. Fix a bijection between N and Q,
which allows us to enumerate all rational numbers as $r_1, r_2, r_3,...$ (this is possible because Q is countable).
Now consider R with the usual Euclidean metric topology and let
$U := \bigcup_{n=1}^{\infty} V_{\frac{1}{2^n}}(r_n)$ and $C := U^c$.
Problem 8. Prove that C is closed.
Problem 9. Prove that C is non-empty. (Hint: suppose it is empty. Then $[0, 10] \subset U$. By compactness of
$[0, 10]$ (why is it compact?), we can select a finite subcover from the open cover \{$V_{\frac{1}{2^n}}(r_n)$\}$_{n \in \mathbb{N}}$. What is the
the sum of the lengths of the intervals in this finite subcover?)
Problem 10. Prove that C is disconnected. (Hint: C should contain at least two elements, between which
there is a rational number.)
