[Author] O. I. Reinov [Title] Geometrical properties of universally measurable mappings [AMS Subj-class] 46B26 Nonseparable Banach spaces [Abstract] It is considered a question under which conditions universally measurable mappings from a Hausdorff topological space $S$ into a metric space $R$ (with a metric $\rho$) belong to the class $\Cal D$ of the following mappings $f: S\to R$: for each compact set $K\subset S$ and every $\varepsilon>0$ there exists an open (in the induced topology) set $V\subset K$ such that the occilation $\omega(f;V)$\ of the $R$-valued function $f$ over $V$ is less than $\varepsilon$ (here $\omega(f;V)=\sup_{s,t\in V} \rho(f(s),f(t))$). [Comments] AmSTeX, 14 pp., Russian [Contact e-mail] orein@orein.usr.pu.ru