View source for Hyperspace

''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 $K(X)$.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,   "Topology" , '''2''' , Acad. Press  (1968)  (Translated from French)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  E. Michael,   "Topologies on spaces of subsets"  ''Trans. Amer. Math. Soc.'' , '''71'''  (1951)  pp. 152–182</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Ponomarev,   "A new space of closed sets and many-valued continuous maps of bicompacts"  ''Mat. Sb.'' , '''48 (90)''' :  2  (1959)  pp. 191–212  (In Russian)</TD></TR>
</table>



====Comments====
The exponential topology is commonly called the Vietoris topology, in honour of L. Vietoris who introduced it in 1922 [[#References|[a1]]]. However, it made little impact on topology for 20 years, until it was rediscovered by O. Frink [[#References|[a2]]] and by E. Michael [[#References|[2]]]. If $X$ is a compact metric space, then the Vietoris topology on $2^X$ is induced by the [[Hausdorff metric]]. For general accounts of hyperspace theory, see [[#References|[a3]]] and [[#References|[a4]]].

On $\text{Exp}_\omega(X)$ one frequently considers the topology generated by the family $\{ [F,O] : F \in \text{Exp}_\omega(X)\,,\ O\,\text{open} \}$, where $[F,O] = \{ G : F \subseteq G \subseteq O \}$. Thus topologized $\text{Exp}_\omega(X)$ is called the Pixley–Roy hyperspace of $X$. It is often used in the construction of counterexamples, see [[#References|[a5]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Vietoris,   "Bereiche zweiter Ordnung"  ''Monatsh. Math. Physik'' , '''32'''  (1922)  pp. 258–280</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Frink,   "Topology in lattices"  ''Trans. Amer. Math. Soc.'' , '''51'''  (1942)  pp. 569–582</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  S.B. Nadler,   "Hyperspaces of sets" , M. Dekker  (1978)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  B.L. McAllister,   "Hyperspaces and multifunctions, the first half century"  ''Nieuw Arch. Wisk. (3)'' , '''26'''  (1978)  pp. 309–329</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  E.K. van Douwen,   "The Pixley–Roy topology on spaces of subsets"  G.M. Reed (ed.) , ''Set-Theoretic Topology'' , Acad. Press  (1977)  pp. 111–134</TD></TR>
</table>

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