 Countable closure under real polynomial roots

Every countable set $B$ of real numbers is contained in a countable set closed under taking all real roots of a polynomial with coefficients in that set. Let $\varphi (A)$ consist of these roots of a polynomial, excluding the zero polynomial. There are countably many finite coefficient lists over a countable set, and each nonzero polynomial has finitely many roots of a polynomial; hence $\varphi (A)$ is countable. Define $B_0=B$ and $B_{n+1}=B_n\cup \varphi (B_n)$. The countable union of countable sets theorem shows that $X={\bigcup }_n{B}_n$ is countable. Every finite coefficient list in $X$ belongs to one stage $B_N$, so its real roots of a polynomial belong to $B_{N+1}\subseteq X$. Retaining $B_n$ at the next stage matters: $\varphi (A)$ need not contain $A$.