Some factorization principles for set-valued functions are presented. In particular the following result is proved: Let $X$ be a topological space and $Y$ be a completely metrizable space. If for every lsc mapping $\theta \:X \to Y$ there exist a completely metrizable space $Z$, a continuous single-valued mapping $g\:Z\to Y$ and a usc mapping $\psi \:X\to Z$ with a perfect property $Q$ such that $g(\psi (x))\subset \theta (x)$ for every $x\in X$, then the same holds for some zero-dimensional space $Z$.