Open maps having the Bula property
Volume 205 / 2009
Fundamenta Mathematicae 205 (2009), 91-104
MSC: Primary 54F45, 54F35, 54C60, 54C65; Secondary 55M10, 54C35, 54B20.
DOI: 10.4064/fm205-2-1
Abstract
An open continuous map $f$ from a space $X$ onto a paracompact $C$-space $Y$ admits two disjoint closed sets $F_0,F_1\subset X$ with $f(F_0)=Y=f(F_1)$, provided all fibers of $f$ are infinite and $C^*$-embedded in $X$. Applications are given to the existence of “disjoint” usco multiselections of set-valued l.s.c. mappings defined on paracompact $C$-spaces, and to special type of factorizations of open continuous maps from metrizable spaces onto paracompact $C$-spaces. This settles several open questions.
