Every open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed sets whose image by f is Y provided all fibers of f are infinite and C-embedded in X. Applications are demonstrated for the existence of disjoint usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and for special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions.