# Hyperspace

(Redirected from Vietoris topology)
over a topological space $X$
A space whose points are the elements of some family $\mathfrak{M}$ of subsets of $X$ with some topology. $\mathfrak{M}$ is usually a ring of sets, but this is not assumed in advance.
Example. $\mathcal{P}(X)$ is the hyperspace of all subsets of a space $X$; a basis for the topology is formed by the sets $\{ M : F \subseteq M \subseteq G \}$ provided that $F$ is closed in $X$, $G$ is open in $X$ and $F \subseteq G$.
The most frequently occurring hyperspace is $2^X$. It consists of all closed subsets of a topological space $X$; a sub-base of the exponential topology on $2^X$ is formed by $\{ F : F \subseteq G \}$ and $\{ F : F \cap H = \emptyset \}$, where $G$ and $H$ are open in $X$ and $F$ runs through $2^X$. Topologies in the following hyperspaces are defined in the same manner: on the set $\mathfrak{S}(X)$ of all compact subsets of a space $X$; on the set $\text{Exp}_\omega(X)$ of all finite subsets of a space $X$; on the set $K(X)$ of all subcontinua (connected compacta) of a continuum $X$, etc. These spaces may be regarded as subspaces of the hyperspace $2^X$ endowed with the exponential topology. If $X$ is a uniform space, then the set $2^X$ has a natural uniform structure; the uniform space obtained in this way is denoted by $H(X)$. If $X$ is compact, then the hyperspaces $2^X$, $\mathfrak{S}(X)$ and $H(X)$ are all homeomorphic and are compact. If $X$ is a compact metrizable space, so is $2^X$. If $X$ is a continuum, so are $2^X$ and $K(X)$.