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Real analysis HW1 Deadline: 10/06 1 In this exercise, we will prove the famous Baire category theorem. Let (X, d) be a complete metric space and (Xn )n∈N be a sequence of closed subsets in X. Assume also that ∀n ∈ N , Int(Xn ) = φ. Then prove that S 1 Int( ∞ n=1 Xn ) = φ. 2 In this exercise, prove that the (standard) Cantor set has uncountable many elements. 2 3 Show that the irrational numbers form a Gδ set, while rational numbers do not. Also, construct a subset in R1 which is neither a Gδ or Fσ set. 1 In fact, there is an equivalent statement which is more concise: countable intersection of open dense subsets of a complete metric space is still dense. In particular, it is nonempty. 2 There are at least two ways to do so. First of all, try to use the Cantor-Lebesgue function we constructed in class. Secondly, you can show that Cantor is perfect, that is, closed and has no isolated point. A consequence followed from Cantor says that perfect set immediately is uncountable. 1