Unformatted text preview: . 6. Probelm # 30 page 46 in Rudin . 7. (a) Show that the Cantor set C = T n ∈ N E n (deﬁned in Rudin 2.44) is uncountable. [ Hint : Use ternary numbers and a diagonal argument similar to Rudin 2.14 .] (b) Show that, for any ± > , there is a union of intervals with total length < ± that contains the Cantor set C . [ Hint : C ⊂ E n , and each of the 2 n intervals in E n is contained in an open interval of length (1 + ± ) / 3 n ]. (c) Show that the Cantor set C ⊂ R is compact. 8. Assume ( X, d ) is a connected metric space. Prove that the only subsets that are both open and closed are X and ∅ . 9. If in a metric space ( X, d ) we have B ⊂ A ⊂ X , then the set B is a connected subset of ( A, d ) (i.e. A with the relative topology) if and only if B is connected subset of ( X, d ) ....
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- Fall '10
- Topology, Metric space, page. Prove Lemma, connected metric space