Exercises for Week 6

Due Wednesday, November 13

  1. Recall that we defined $\mathbb{Q}$ as the quotient $(\mathbb{Z}\times (\mathbb{Z} \setminus \{0\})) / \mathord{\sim}$ where $(a,b) \sim (c,d) \iff ad = bc$. Define appropriate operations of addition, multiplication, negation, and multiplicative inverse for $\mathbb{Q}$. For all but addition, you may assert that your operation is well-defined on equivalence classes without checking this.
  2. Prove that $\aleph_1$ is the least ordinal that does not admit an order-preserving function into $\mathbb{R}$ (with respect to the usual strict orderings on the ordinals and the reals.)
  3. Prove that if $G$ is an intersection of countably many dense open subsets of $\mathbb{R}$, then there is an injection $2^\omega \to G$. Hint: use ideas from the proof of the Baire Category Theorem and the proof that every perfect set of reals admits an injection from $2^\omega$.