I am trying to read Michael Hrusák's ["MAD Families and the rationals"](http://www.matmor.unam.mx/~michael/reprints_files/MAD_families_and_the_rationals.pdf) and I have studied Forcing using Kunen's books. On theorem 1, the author says that no new real is contained in a countable closed set coded in the ground model. If I understood it right, what the author stated is that if $M$ is a ctm and $A\subset (2^\omega)^M$ is such that $A \in M$ and $\operatorname{cl}^MA$ is countable , then $\operatorname{cl}^{M[G]}A=\operatorname{cl}^{M}A$. My question is: Why?



No new real is contained in a countable closed set
While you can code a closed set $A \subseteq 2^\omega$ by any countable dense subset of $A$; I think it is more transparent to code $A$ by the tree $T\_A:=\{ f\upharpoonright n : f\in A, n\in \omega\}$. Then $A=[T\_A] = \{f: \forall n\ f\upharpoonright n \in T\}$. 


Any countable closed set $A$ can be decomposed by Cantor-Bendixson derivatives as follows: 


* Let $T\_0 = T\_{A}$, $A\_0 = A$, and let $B\_0$ be the set of isolated points of $A\_0$ (which can be found as the isolated branches of $T\_0$).
* Set $T\_1 = $ the nodes of $T\_0$ which have at least two extensions in $A\_0$, and $A\_1 = A\_0 \setminus B\_0$.
* Then $A\_1 = [T\_1]$ = the set of non-isolated points of $A$ = the "first CB derivative" of $A$.
* Define $A\_2$, $T\_2$ similarly. etc.
* Let $T\_\omega$ be the intersection of the decreasing sequence $T\_n$, and continue by letting $T\_{\omega+1}$ be the derivative of $T\_\omega$.
* Continue by transfinite induction. Eventually (after countably many steps) you will reach the empty set.
* You have now written $A$ as the disjoint union of set $B\_0 \cup B\_1\cup \cdots$.


The construction of the sequence $T\_\alpha$ is absolute (as the definition off the derivative is arithmetical, it used only quantifications over the countable set $2^{<\omega}$). So it will give the same results in any model. 


If you compare $V$ and $V[G]$, the sets $B\_0$ defined in these models (isolated branches of $T\_0$) will be the same. But then also the sets $B\_1$ etc will be the same in both models. etc. 


