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Real analysis HW1
Deadline: 10/06
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
Int( ∞
n=1 Xn ) = φ.
In this exercise, prove that the (standard) Cantor set has uncountable many
elements. 2
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.
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.
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.