2013/11/03 by Vesko Valov, Valov, Vesko
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Equations Stability Results #General Topology (math.GN) #Primary 54C65 #Secondary 54F65 #math.GN #msc:54C65 #msc:54F65
paper · pdf · doi:10.48550/arxiv.1311.0476
8 pages
arxiv created 2013/11/03 · openalex publication_date 2013/11/03 · arxiv updated 2013/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The following selection theorem is established: Let X be a compactum possessing a binary normal subbase \mathcal S for its closed subsets. Then every set-valued \mathcal S-continuous map Φ\colon Z→ X with closed \mathcal S-convex values, where Z is an arbitrary space, has a continuous single-valued selection. More generally, if A⊂ Z is closed and any map from A to X is continuously extendable to a map from Z to X, then every selection for Φ|A can be extended to a selection for Φ. This theorem implies that if X is a κ-metrizable (resp., κ-metrizable and connected) compactum with a normal binary closed subbase \mathcal S, then every open \mathcal S-convex surjection f\colon X→ Y is a zero-soft (resp., soft) map. Our results provide some generalizations and specifications of Ivanov's results (see \citei1, \citei2, \citei3) concerning superextensions of κ-metrizable compacta.