Let $\mathbb C$ be the complex plane, $H(\mathbb C)$ the set of all entire functions, and $D(\mathbb C)$ the set of all non-negative divisors in $\mathbb C$.

Consider the map $Z:H(\mathbb C)\to D(\mathbb C)$ which to every entire function $f\in H(\mathbb C)$ puts into correspondence its divisor of zeros. There are natural topologies which make this map continuous: on $H(\mathbb C)$ this is uniform convergence on compact subsets, and on $D(\mathbb C)$ the induced topology of weak convergence of measures. (A divisor can be thought of as a discrete measure that takes integer values on all bounded sets).

The Weierstrass factoriation theorem says that this map is surjective, and he actually constructed a right inverse $W$, so that $Z\circ W=id_{D(\mathbb C)}$.

Question: Does there exist a CONTINUOUS right inverse?

Weierstrass map is evidently not continuous. (His map depends on the ordering of zeros by absolute value, and this ordering can change when the divisor varies continuously). I can prove that there is no analytic right inverse, with the natural analytic structures on $H(\mathbb C)$ and $D(\mathbb C)$. I can also prove that there is no multiplicative continuous right inverse (multiplicative means that the sum of divisors corresponds to the product of functions).

EDIT. Let me add for completeness a brief explanation why there is no analytic $W$. In MR2280501, Michigan Math. J. 54 (2006), no. 3, 687–696, for every compact $E\in U$ of zero log capacity in the unit disk $U$, I constructed an holomrphic function $F(z,w)$ of two variables $(z,w)\in\mathbb C\times U$, with the property that $F(z,w)\neq 0$ when $w\in E, z\in\mathbb C$ but $z\mapsto F(z,w)$ has zeros for $w\in U\backslash E$. Take some uncountable $E$. Let $D(w)$ be the divisor of the entire function $f_w=F(.,w)$ in the $z$-plane. Assuming that an analytic Weierstrass map exists, denote $g=W(\emptyset)$. Then the set $\{ w:W(D(w))=g\}$ must be analytic (that is either discrete of the whole $w$-disk, but this is not so because it contains $F$ and is not equal to the whole disk.