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I've attempted to write an arbitrary subset of real numbers as the union of it's single points.

Any single point set is closed. But I don't know if the union of all single points of a subset of reals like A is closed or not? (suppose A is infinite) also i have this problem for writing A as the intersect of open subsets, because the intersection of an infinite number of open subsets may not be open. for example can we write [0,1) as the union of the sets [0, 1 - 1/n] for n in naturals, as the union of countable infinite closed sets?

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  • $\begingroup$ The union of singletons in $A$ writes an arbitrary set $A$ as a union of closed sets, as you say. For the other part, think about complements... $\endgroup$ – Ned Dec 18 '18 at 15:09
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Hint

$A= \bigcup_{x\in A} \{x\}$

Also

$A= \bigcap_{x\in A^c} \mathbb{R}-\{x\}$

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