Exercises for Week 6
Due Wednesday, November 13
- 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.
- 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.)
- 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 $2^\omega$.