Finite graphs have unique second and third symmetric product suspension

Abstract

Let X be a metric continuum and n be a positive integer. Consider the hyperspace Fn(X) of all nonempty closed subsets of X with at most n points. Given n≥2, the n-fold symmetric product suspension of X is the quotient space Fn(X)/F1(X) which is obtained from Fn(X) by identifying F1(X) into a one-point set. In this paper we prove that if n∈{2,3}, X is a finite graph, and Y is a continuum such that Fn(X)/F1(X) is homeomorphic to Fn(Y)/F1(Y), then X is homeomorphic to Y. This result completes the work of Germán Montero-Rodríguez, David Herrera-Carrasco, María de J. López and Fernando Macías-Romero who previously proved the corresponding result, for each n≥4.