Open mappings looking like projections

Every open continuous mappingf from a metric space (X, d) onto a countable-dimensional metric spaceY admits a special type of factorization (Y×0, 1] throughout), provided all fibers off are dense in itself and complete with respect tod. On this basis, an upper semi-continuous Cantor bouquet of disjoint usco selections for a class of 1.s.c. mappings between metrizable spaces is constructed.