set theory?

Question

1. Show how to write the set of irrationals as a countable intersection of cofinite subsets of R.

2. Let A be an uncountable subset of R.
Prove that there is a subset of distinct elements {a_n : n=1,2,...} of A such that sum of a_n (from n=1 to infinity) diverges.

Answer 1

1. Q (the rationals) is countable. For each q in Q, define the set
A(q) = {r in R : r does not equal q}, so A(q) is the complement of the single point (finite) set {q}.
Then the set U = (union(over q in Q) of A(q)) is a countable union of cofinite sets.


2. For each positive integer n, define the set
B(n) = {r in R : r>(1/n)}
and for each negative integer n define the set
C(n) = {r in R : r<(1/n)}.
Then R = (union(over pos. int.'s n) of B(n)) union (union(over negative integers n) of C(n)) union {0}.
A is contained in this countable union, so at least one of the sets B(n) or C(n) must contain infinitely many elements of A because A is uncountable (If they all contained finitely many elements of A then A would be a countable union of finite sets, and would therefore be countable).
Without loss of generality, suppose B(n) contains infinitely many elements of A. Then choose a sequence {a(k)} from that infinite set of elements of A and their sum is
a(1) + a(2) + a(3) + ... >= (1/n) + (1/n) + (1/n) + ... = infinity

Answer 2

1. Huh??????? The irrationals are an uncountable set so it's gonna be difficult to get them from a countable intersection ☺

2. This falls right out of the definition of an uncountable subset. You might want to think about taking (or reviewing *really* deeply) a course in Analysis before you take Topology ☺


Doug