# Prove that we can write any subset of Real numbers as a Union of closed subsets, or intersect of open subsets.

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?

• 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... – Ned Dec 18 '18 at 15:09

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