Let $\lambda\to F(\lambda)$ a family of meromorphic functions on an open set $\Omega$ of $\mathbb{C}$ and let $\mu\in \mathbb{C}$.
Put $U_\mu=\{\lambda\in \Omega: \mu$ is not a pole of $ F(\lambda) \}$.
My question: is $U_\mu$ an open set of $\mathbb{C}$?
No, not without further restrictions. Right now there is absolutely no condition on how this family depends on $\lambda$.
For example let $F(\lambda)(z)=\begin{cases}\frac1z &\lambda\neq0\\z&\lambda=0\end{cases}$ , then $U_0=\{0\}$ is not open.