2024/01/25 by Valentin Gutev, Gutev, Valentin
Computer Science · Mathematics · #54A20 #54B20 #54C65 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2401.14384
openalex publication_date 2024/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If f is a continuous selection for the Vietoris hyperspace \mathscrF(X) of the nonempty closed subsets of a space X, then the point f(X)∈ X is not as arbitrary as it might seem at first glance. In this paper, we will characterise these points by local properties at them. Briefly, we will show that p=f(X) is a strong butterfly point precisely when it has a countable clopen base in U for some open set U⊂ X∖\p\ with U=U∪\p\. Moreover, the same is valid when X is totally disconnected at p=f(X) and p is only assumed to be a butterfly point. This gives the complete affirmative solution to a question raised previously by the author. Finally, when p=f(X) lacks the above local base-like property, we will show that \mathscrF(X) has a continuous selection h with the stronger property that h(S)=p for every closed S⊂ X with p∈ S.