The main result of this note is the following common generalization of a theorem of J.-P. Vigu\'e on selfmaps of bounded convex domains and a classical Hurwitz's theorem: Let $X$ be a complex submanifold of a complex manifold $Y$ and let $f_n$ be a sequence in $\mathcal{H}(X, Y)$ converging to $f \in \mathcal{H}(X, Y)$ with $ X \cap\limsup \hbox{Fix} (f_n) = \emptyset$, where $\mathcal{H}(X, Y)$ represents the space of holomorphic maps from $X$ to $Y$ with the compact-open topology and $\hbox{Fix}(f)$ denotes the set of fixed points of $f$. Then either $\hbox{Fix}(f)=\emptyset$ or $\dim \hbox{Fix}(f) \geq 1$. \smallskip In addition, properties of fixed point sets of selfmaps of bounded convex domains discovered by Vigu\'e - for example, that such sets are holomorphic retracts - are extended to other normal maps.