Topology of the complement of a countable union of divisors I have the following situation: let $U\subset\mathbb{P}^n$ ($n\geq2$) be an open subset and let $U'$ the complement of a countably infinite union of divisors in $U$. 
Is there any way to know the relation between $\pi_1(U')$ and $\pi_1(U)$?
For example I was wondering if the induced map $\pi_1(U')\to\pi_1(U)$ was surjective. Notice that here we are working over $\mathbb{C}$ hence a divisor has real codimension $2$.
A theorem of Godbillon (cf. Francesco's answer here) states that we have surjectivity in the case of a unique divisor.
How can we pass from this to a countable union? Is there any topological spectral sequence which can handle this question?
References are much appreciated.
Thank you very much.
 A: Here is a proof of surjectivity. (I am assuming that $U$ is open and connected in the classical topology.) First of all, equip $U$ with a complete Riemannian metric $g$ and let $d$ denote the associated distance function.  (You obtain $g$ by multiplying your favorite Riemannian metric on ${\mathbb C}P^n$ by a positive scalar function on $U$ which diverges to $\infty$ at least quadratically as you approach the boundary of $U$.)  Next, given a loop $L$ in $U$ (all loops will be based at a point $x\in U'$) you can approximate $L$ by a polygonal loop $P$ with $k$ vertices so that $P$ represents the same based homotopy class as $L$. (Here "polygonal" can be defined, for instance using circular arcs contained in projective lines in ${\mathbb C}P^n$.) Let $Pol$ denote the space of polygonal loops in $U$ based at $x$ with $k$ vertices and base-homotopic to $L$. I will equip $Pol$ with the Hausdorff distance defined via the metric $d$ on $U$. I will leave it to you to prove that  $Pol$  is a complete metric space. It follows from Sard's theorem or from Kleiman transversality, whatever you prefer, that for each divisor $D$ in ${\mathbb C}P^n$, the subset $Pol(D)$ consisting of loops in $Pol$ which are disjoint from $D$, is open and dense. Now, given a countable collection $\{D_j: j\in J\}$ of divisors, it follows from the Baire category theorem (applied to $Pol$) that the intersection
$$
\bigcap_{j\in J} Pol(D_j)
$$
is still dense in $Pol$. Hence, there exists a polygonal loop $P$ based at $x$ and contained in 
$$
U'= U - \bigcup_{j} D_j
$$ 
such that $P$ represents the same element of $\pi_1(U,x)$ as $L$. Thus, $\pi_1(U',x)\to \pi_1(U,x)$ is surjective. 
