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# Countable and (un)countable

Find an example of a set $$S$$ such that $$S$$ is countable, yet $$S'$$ (the set of all limit points of $$S$$) is uncountable.

Note by Austin Stromme
3 years, 5 months ago

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The rationals. More generally, if $$A$$ is an uncountable compact subset of a metric space then, for any $$n \in \mathbb{N}$$ there is a finite subset $$S_n$$ of $$A$$ such that every element of $$A$$ is within $$n^{-1}$$ of some element of $$S_n$$. Then $S=\bigcup_{n\in\mathbb{N}}S_n$ Is a countable subset of $$A$$ whose closure is $$A$$.

- 3 years, 5 months ago

One example is the set $$S:= \{ x \colon x\in \mathbb{R}^2 \land \| x\| <1 \}\cap \mathbb{Q}^2$$.

- 3 years, 5 months ago