Let $f : \Lambda \rightarrow X$ be a continuous surjective map, where $\Lambda$ is a complex manifold and $X$ a topological space. Suppose that for all $x \in X$, there is a neighborhood $U_x$ of $X$ and a continuous map : $g : U_x \rightarrow \Lambda$ which is a right inverse of $f$.
Does this always allow one to define a complex atlas on $X$ ? More precisely, is it necessarily true in this generality that the transition maps $g_1 \circ g_2^{-1}=g_1 \circ f$ are holomorphic, when $g_i$ are local sections of $f$ defined on a common open set in $X$ ?