Paper 4, Section II, E

(a) What is a countable set? Let $X, A, B$ be sets with $A, B$ countable. Show that if $f: X \rightarrow A \times B$ is an injection then $X$ is countable. Deduce that $\mathbb{Z}$ and $\mathbb{Q}$ are countable. Show too that a countable union of countable sets is countable.

(b) Show that, in the plane, any collection of pairwise disjoint circles with rational radius is countable.

(c) A lollipop is any subset of the plane obtained by translating, rotating and scaling (by any factor $\lambda>0$ ) the set

$\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\} \cup\left\{(0, y) \in \mathbb{R}^{2} \mid-3 \leqslant y \leqslant-1\right\} .$

What happens if in part (b) we replace 'circles with rational radius' by 'lollipops'?

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