Open maps having the Bula property

TL;DR

This paper proves that under certain conditions, open maps from a space onto a paracompact C-space have disjoint closed subsets whose images cover the entire target space, with applications to set-valued mappings and factorizations.

Contribution

It establishes the Bula property for open maps with infinite fibers onto paracompact C-spaces, answering several open questions in topology.

Findings

Existence of disjoint closed subsets with full image coverage.

Applications to set-valued multiselections of l.s.c. mappings.

Factorizations of open maps from metrizable spaces.

Abstract

Every open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed subsets of X so that their 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.