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

[1]	K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French)
[2]	E. Michael, "Topologies on spaces of subsets" Trans. Amer. Math. Soc. , 71 (1951) pp. 152–182
[3]	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)

Comments

The exponential topology is commonly called the Vietoris topology, in honour of L. Vietoris who introduced it in 1922 [a1]. However, it made little impact on topology for 20 years, until it was rediscovered by O. Frink [a2] and by E. Michael [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 [a3] and [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 [a5].

References

[a1]	L. Vietoris, "Bereiche zweiter Ordnung" Monatsh. Math. Physik , 32 (1922) pp. 258–280
[a2]	O. Frink, "Topology in lattices" Trans. Amer. Math. Soc. , 51 (1942) pp. 569–582
[a3]	S.B. Nadler, "Hyperspaces of sets" , M. Dekker (1978)
[a4]	B.L. McAllister, "Hyperspaces and multifunctions, the first half century" Nieuw Arch. Wisk. (3) , 26 (1978) pp. 309–329
[a5]	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