Dendrites and symmetric products

TL;DR

This paper investigates the topological properties of hyperspaces of dendrites, showing that homeomorphic hyperspaces imply the underlying spaces are also dendrites with closed end point sets.

Contribution

It establishes a topological invariance result for hyperspaces of dendrites, linking the homeomorphism of hyperspaces to the dendritic structure of the original spaces.

Findings

Homeomorphic hyperspaces imply the underlying spaces are dendrites.

Dendrites with closed end point sets are characterized by their hyperspaces.

The result applies to hyperspaces of up to n points with the Hausdorff metric.

Abstract

For a given continuum $X$ and a natural number $n,$ we consider the hyperspace $F_n(X)$ of all nonempty subsets of $X$ with at most $n$ points, metrized by the Hausdorff metric. In this paper we show that if $X$ is a dendrite whose set of end points is closed, $n \in \mathbb{N}$ and $Y$ is a continuum such that the hyperspaces $F_n(X)$ and $F_n(Y)$ are homeomorphic, then $Y$ is a dendrite whose set of end points is closed.