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''over a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485101.png" />''
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''over a topological space $X$''
  
A space whose points are the elements of some family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485102.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485103.png" /> with some topology. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485104.png" /> is usually a ring of sets, but this is not assumed in advance.
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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. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485105.png" /> is the hyperspace of all subsets of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485106.png" />; a basis for the topology is formed by the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485107.png" /> provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485108.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h0485109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851010.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851012.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851013.png" />. It consists of all closed subsets of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851014.png" />; a sub-base of the [[Exponential topology|exponential topology]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851015.png" /> is formed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851019.png" /> are open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851021.png" /> runs through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851022.png" />. Topologies in the following hyperspaces are defined in the same manner: on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851023.png" /> of all compact subsets of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851024.png" />; on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851025.png" /> of all finite subsets of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851026.png" />; on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851027.png" /> of all subcontinua (connected compacta) of a continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851028.png" />, etc. These spaces may be regarded as subspaces of the hyperspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851029.png" /> endowed with the exponential topology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851030.png" /> is a [[Uniform space|uniform space]], then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851031.png" /> has a natural uniform structure; the uniform space obtained in this way is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851033.png" /> is compact, then the hyperspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851036.png" /> are all homeomorphic and are compact. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851037.png" /> is a compact metrizable space, so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851039.png" /> is a continuum, so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851041.png" />.
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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====
 
====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>
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<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>
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<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>
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</table>
  
  
  
 
====Comments====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851042.png" /> is a compact metric space, then the Vietoris topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851043.png" /> is induced by the [[Hausdorff metric|Hausdorff metric]]. For general accounts of hyperspace theory, see [[#References|[a3]]] and [[#References|[a4]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851044.png" /> one frequently considers the topology generated by the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851046.png" />. Thus topologized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851047.png" /> is called the Pixley–Roy hyperspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048510/h04851048.png" />. It is often used in the construction of counterexamples, see [[#References|[a5]]].
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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====
 
====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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<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>
 +
 
 +
{{TEX|done}}

Revision as of 16:29, 16 April 2016

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
How to Cite This Entry:
Hyperspace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperspace&oldid=38576
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article