A note on hyperspaces by closed sets with Vietoris topology

2021/11/21 by Liu, Chuan, Lin, Fucai
#54B20 #54D20 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2111.10710

Abstract

For a topological space X, let CL(X) be the set of all non-empty closed subset of X, and denote the set CL(X) with the Vietoris topology by (CL(X), \mathbbV). In this paper, we mainly discuss the hyperspace (CL(X), \mathbbV) when X is an infinite countable discrete space. As an application, we first prove that the hyperspace with the Vietoris topology on an infinite countable discrete space contains a closed copy of n-th power of Sorgenfrey line for each n∈ℕ. Then we investigate the tightness of the hyperspace (CL(X), \mathbbV), and prove that the tightness of (CL(X), \mathbbV) is equal to the set-tightness of X. Moreover, we extend some results about the generalized metric properties on the hyperspace (CL(X), \mathbbV). Finally, we give a characterization of X such that (CL(X), \mathbbV) is a γ-space.

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