Paper 4 , Section II, E

What does it mean for a set to be countable ?

Show that $\mathbb{Q}$ is countable, but $\mathbb{R}$ is not. Show also that the union of two countable sets is countable.

A subset $A$ of $\mathbb{R}$ has the property that, given $\epsilon>0$ and $x \in \mathbb{R}$, there exist reals $a, b$ with $a \in A$ and $b \notin A$ with $|x-a|<\epsilon$ and $|x-b|<\epsilon$. Can $A$ be countable ? Can $A$ be uncountable ? Justify your answers.

A subset $B$ of $\mathbb{R}$ has the property that given $b \in B$ there exists $\epsilon>0$ such that if $0<|b-x|<\epsilon$ for some $x \in \mathbb{R}$, then $x \notin B$. Is $B$ countable ? Justify your answer.

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