Intersection of countable collection of closed intervals

Question

Is the following statement true?

$A\subseteq\mathbb{R}$ closed $\iff$ A is the intersection of a countable collection of closed intervals

For $\impliedby$:

Suppose A is the intersection of a countable collection of closed intervals. The intersection of finite or infinite number of closed sets is closed. So A must be closed.

For $\implies$:

If A is closed, then it can be written as the intersection of arbitrarily many closed intervals. I am not sure if this means that the collection is countable though.

Answer 1

The intersection of any collection of closed intervals is either empty or a closed interval. (Just verify that if two points belong to the intersection then any point between these two is also in the intersection). Hence the stated result is false.