arXiv subjects
Publications and source records attributed to Valentin Gutev.
We demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve this reduction is also applied to get a simple direct proof of that selection theorem of Michael's. Some other possible applications are demonstrated as well.
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.
The present paper improves a result of V. Gutev and T. Nogura (1999) showing that a space $X$ is topologically well-orderable if and only if there exists a selection for $\mathcal{F}_2(X)$ which is continuous with respect to the Fell topology on $\mathcal{F}_2(X)$. In particular, this implies that $\mathcal{F}(X)$ has a Fell-continuous selection if and only if $\mathcal{F}_2(X)$ has a Fell-continuous selection.