MODELS AND HOMOGENEITY DEGREE OF HYPERSPACES OF A SIMPLE CLOSED CURVE

Abstract

Given a continuum $X$ and $n\in \mathbb{N}$, let $C_n(X)$ (resp., $F_n(X)$) be the hyperspace of nonempty closed sets with at most $n$ components (resp., $n$ points). Let $S^1$ denote the unit circle in the plane. Given $1\le m\le n$, we consider the quotient space $C_n(S^1)∕F_m(S^1)$. The homogeneity degree of $X$, $\text{hd}(X)$, is the number of orbits of the group of homeomorphisms of $X$. We discuss the known models for the hyperspaces of $S^1$, we construct a new model for a hyperspace of $S^1$ by proving that $C_2(S^1)∕F_2(S^1)$ is homeomorphic to the topological suspension of a solid torus, and we show that $\text{hd}(C_2(X)∕F_2(X))=3$ and $\text{hd}(C_2(X)∕F_1(X))=4$.