Convex sets, half-spaces and countable intersections

Question

Is it possible to proof that in $\mathbb{R}^{n}$ all convex and closed sets can be written as a countable intersection of half-spaces.
In my opinion the disk in $\mathbb{R}^{2}$ is already a counter-example, but I don't know how to prove it. Moreover, the fact that $\mathbb{Q}$ is countable and dense in $\mathbb{R}$ makes me feel that maybe the statement is correct.

Answer 1

Let $C \subset E$ be a convex set in $E= \bf R^n$ and $C'\subset E^*$ the set of linear form such that $C \subset \{ f\leq 1\}$. it is well known that if $C$ is closed, $C$ is the intersection $\cap _{f\in C'} \{ f\leq 1\}$. So choosing a dense subset in $C'$ one gets the result, under the assumption that $C$ is closed.