Is the convex hull of a countable set a Borel set?

Question

The convex hull of a subset $X$ of $\mathbb{R}^2$ is the smallest convex subset of $\mathbb{R}^2$ containing $X$. My question is, if $X$ is countable, then is the convex hull of $X$ necessarily a Borel set?

If not, does anyone know of a counterexample?

Answer 1

Yes，it is Borel.

Define $\text{conv}_n(E)$ by $x\in \text{conv}_n(E)$ iff $\exists a_1,a_2,\dots,a_n\in E,\exists t_1,t_2,\dots,t_n\in[0,1], \Sigma t_i=1$ such that $x=\Sigma_{i=1}^n t_ia_i$.

It is easily to seen that $\text{conv}_n(E)$ is Borel, so $\text{conv}(E)$ is Borel since $\text{conv}_n(E)=\cup_{n=1}^\infty\text{conv}_n(E)$ (Try to prove this equality,please note that the union is vonvex and contains $E$, or refer J. van Mill's book-infinite-Dimensional Topology, Page 7,Lemma 1.2.2)