April 2023 HOMOGENEITY DEGREE OF HYPERSPACES OF ARCS AND SIMPLE CLOSED CURVES
Rodrigo Hernández-Gutiérrez, Alejandro Illanes, Verónica Martínez-de-la-Vega
Rocky Mountain J. Math. 53(2): 463-476 (April 2023). DOI: 10.1216/rmj.2023.53.463

Abstract

Given a continuum X and n, let Cn(X) (resp. Fn(X)) be the hyperspace of nonempty closed sets with at most n components (resp. n points). Given 1mn, we consider the quotient space Cn(X)Fm(X). The homogeneity degree of X, Hd(X), is the number of orbits of the group of homeomorphisms of X. We discuss lower bounds for the homogeneity degree of the hyperspaces Cn(X), Cn(X)Fm(X) when X is a finite graph. In particular, we prove that for a finite graph X:

  • (a) Hd(Cn(X)Fm(X))=1 if and only if X is a simple closed curve and n=m=1,

  • (b) Hd(Cn(X)Fm(X))=2 if and only if X is an arc and either n=m=1 or n=2 and m{1,2},

  • (c) Hd(Cn(X)Fm(X))=3 if and only if X is a simple closed curve and n=m=2, and

  • (d) Hd(Cn(X)Fm(X))=4 if and only if X is a simple closed curve, n=2 and m=1.

Citation

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Rodrigo Hernández-Gutiérrez. Alejandro Illanes. Verónica Martínez-de-la-Vega. "HOMOGENEITY DEGREE OF HYPERSPACES OF ARCS AND SIMPLE CLOSED CURVES." Rocky Mountain J. Math. 53 (2) 463 - 476, April 2023. https://doi.org/10.1216/rmj.2023.53.463

Information

Received: 1 February 2022; Revised: 8 April 2022; Accepted: 20 June 2022; Published: April 2023
First available in Project Euclid: 20 June 2023

MathSciNet: MR4604767
Digital Object Identifier: 10.1216/rmj.2023.53.463

Subjects:
Primary: 54F16
Secondary: 54B15 , 54F50

Keywords: arc , continuum , finite graph , homogeneity , homogeneity degree , hyperspace , hyperspace suspension , Orbit , simple closed curve

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

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Vol.53 • No. 2 • April 2023
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