A continuum is a nondegenerate compact connected metric space. Given a continuum $X$, the most known hyperspaces are:
$$2^X = \{A\subset X : A is nonempty and closed},$$
$$ C_n(X) = \{A\subset 2^X : A has at most n components\},$$ and
$$F_n(X)=\{A\subset2^X :A has at most n points\}.$$
A model for a hyperspace $\mathcal H(X) \in \{2^X , C_n(X), F_n(X)\}$, is a more familiar continuum $Z$ such that $Z$ is homeomorphic to $mathcal H(X)$, and in most of the cases $Z$ is a subset of some Euclidean space whose elements are points instead of subsets of a continuum.
For some specific continua $X$, sometimes it is possible to construct models for either $2^X$ , $C_n(X)$ or $F_n(X)$. In this talk we will present all the known models for hyperspaces. |