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The weakest topology on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369401.png" /> of all closed subsets of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369402.png" /> in which the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369403.png" /> are open (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369404.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369405.png" /> is open, and closed (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369406.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369407.png" /> is closed. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369408.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e0369409.png" /> denotes the set of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694010.png" /> that are closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694011.png" />.
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Example. The topology of the metric space of closed bounded subsets of a metric space endowed with the [[Hausdorff metric|Hausdorff metric]]. The general definition is: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694012.png" /> be an arbitrary finite collection of non-empty open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694013.png" />; a basis for the exponential topology consists of sets of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694014.png" /></td> </tr></table>
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The weakest topology on the set  $  \mathop{\rm exp}  X = 2  ^ {X} $
 +
of all closed subsets of a topological space  $  X $
 +
in which the sets  $  \mathop{\rm exp}  A $
 +
are open (in  $  \mathop{\rm exp}  X $)
 +
if  $  A $
 +
is open, and closed (in  $  \mathop{\rm exp}  X $)
 +
if  $  A $
 +
is closed. If  $  A \subseteq X $,
 +
then  $  \mathop{\rm exp}  A $
 +
denotes the set of all subsets of  $  A $
 +
that are closed in  $  X $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694015.png" /></td> </tr></table>
+
Example. The topology of the metric space of closed bounded subsets of a metric space endowed with the [[Hausdorff metric|Hausdorff metric]]. The general definition is: Let  $  U _ {1} \dots U _ {n} $
 +
be an arbitrary finite collection of non-empty open sets in  $  X $;  
 +
a basis for the exponential topology consists of sets of the form
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694016.png" /> denotes the point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694017.png" /> corresponding to a given closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694018.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694019.png" /> endowed with the exponential topology is called the exponent of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694021.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694022.png" />-space, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694024.png" /> is regular, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694025.png" /> is a Hausdorff space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694026.png" /> is normal, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694027.png" /> is completely regular. For the exponential topology normality is equivalent to compactness. If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694028.png" /> is compact, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694029.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694030.png" /> is a dyadic compactum and the weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694031.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694033.png" /> is also a dyadic compactum. On the other hand, the exponent of any compactum of weight greater than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694034.png" /> is not a dyadic compactum. The exponent of a Peano continuum is an absolute retract in the class of metric compacta and, consequently, it is a continuous image of an interval. However, an exponent of uncountable weight is not a continuous image of the Tikhonov cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694036.png" /> be a closed mapping of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694037.png" /> onto a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694038.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694039.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694040.png" /> is called the exponential mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694041.png" /> is a continuous mapping of a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694042.png" /> onto a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694043.png" />, then it is open if and only if the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694044.png" /> is open. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694045.png" /> acting from the category of compacta and continuous mappings into the same category is a covariant functor of exponential type. Here to a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694046.png" /> there corresponds its exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036940/e03694047.png" />.
+
$$
 +
\langle  U _ {1} \dots U _ {n} \rangle =
 +
$$
 +
 
 +
$$
 +
= \
 +
\left \{ \widehat{F}  \in  \mathop{\rm exp}  X :  F \subseteq \cup _ { 1 } ^ { n }
 +
U _ {i}  \&  F \cap U _ {i} \neq \emptyset , i = 1 \dots n \right \} ,
 +
$$
 +
 
 +
where  $  \widehat{F}  $
 +
denotes the point of $  \mathop{\rm exp}  X $
 +
corresponding to a given closed set $  F \subseteq X $.  
 +
The space $  \mathop{\rm exp}  X $
 +
endowed with the exponential topology is called the exponent of the space $  X $.  
 +
If $  X $
 +
is a $  T _ {1} $-
 +
space, then so is $  \mathop{\rm exp}  X $.  
 +
If $  X $
 +
is regular, then $  \mathop{\rm exp}  X $
 +
is a Hausdorff space. If $  X $
 +
is normal, then $  \mathop{\rm exp}  X $
 +
is completely regular. For the exponential topology normality is equivalent to compactness. If the space $  X $
 +
is compact, then so is $  \mathop{\rm exp}  X $.  
 +
If $  X $
 +
is a dyadic compactum and the weight of $  X $
 +
does not exceed $  \aleph _ {1} $,  
 +
then $  \mathop{\rm exp}  X $
 +
is also a dyadic compactum. On the other hand, the exponent of any compactum of weight greater than or equal to $  \aleph _ {2} $
 +
is not a dyadic compactum. The exponent of a Peano continuum is an absolute retract in the class of metric compacta and, consequently, it is a continuous image of an interval. However, an exponent of uncountable weight is not a continuous image of the Tikhonov cube $  I  ^  \tau  $.  
 +
Let $  f : X \rightarrow Y $
 +
be a closed mapping of a space $  X $
 +
onto a space $  T $.  
 +
The mapping $  \mathop{\rm exp}  f :   \mathop{\rm exp}  X \rightarrow  \mathop{\rm exp}  Y $
 +
defined by $  (  \mathop{\rm exp}  f  ) ( \widehat{F}  ) = ( f ( F) ) \widehat{ {}}  $
 +
is called the exponential mapping. If $  f : X \rightarrow Y $
 +
is a continuous mapping of a compactum $  X $
 +
onto a compactum $  Y $,  
 +
then it is open if and only if the mapping $  \mathop{\rm exp}  f $
 +
is open. The functor $  \mathop{\rm exp}  X $
 +
acting from the category of compacta and continuous mappings into the same category is a covariant functor of exponential type. Here to a morphism $  f $
 +
there corresponds its exponent $  \mathop{\rm exp}  f $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1–2''' , Acad. Press  (1966–1968)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1–2''' , Acad. Press  (1966–1968)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:38, 5 June 2020


The weakest topology on the set $ \mathop{\rm exp} X = 2 ^ {X} $ of all closed subsets of a topological space $ X $ in which the sets $ \mathop{\rm exp} A $ are open (in $ \mathop{\rm exp} X $) if $ A $ is open, and closed (in $ \mathop{\rm exp} X $) if $ A $ is closed. If $ A \subseteq X $, then $ \mathop{\rm exp} A $ denotes the set of all subsets of $ A $ that are closed in $ X $.

Example. The topology of the metric space of closed bounded subsets of a metric space endowed with the Hausdorff metric. The general definition is: Let $ U _ {1} \dots U _ {n} $ be an arbitrary finite collection of non-empty open sets in $ X $; a basis for the exponential topology consists of sets of the form

$$ \langle U _ {1} \dots U _ {n} \rangle = $$

$$ = \ \left \{ \widehat{F} \in \mathop{\rm exp} X : F \subseteq \cup _ { 1 } ^ { n } U _ {i} \& F \cap U _ {i} \neq \emptyset , i = 1 \dots n \right \} , $$

where $ \widehat{F} $ denotes the point of $ \mathop{\rm exp} X $ corresponding to a given closed set $ F \subseteq X $. The space $ \mathop{\rm exp} X $ endowed with the exponential topology is called the exponent of the space $ X $. If $ X $ is a $ T _ {1} $- space, then so is $ \mathop{\rm exp} X $. If $ X $ is regular, then $ \mathop{\rm exp} X $ is a Hausdorff space. If $ X $ is normal, then $ \mathop{\rm exp} X $ is completely regular. For the exponential topology normality is equivalent to compactness. If the space $ X $ is compact, then so is $ \mathop{\rm exp} X $. If $ X $ is a dyadic compactum and the weight of $ X $ does not exceed $ \aleph _ {1} $, then $ \mathop{\rm exp} X $ is also a dyadic compactum. On the other hand, the exponent of any compactum of weight greater than or equal to $ \aleph _ {2} $ is not a dyadic compactum. The exponent of a Peano continuum is an absolute retract in the class of metric compacta and, consequently, it is a continuous image of an interval. However, an exponent of uncountable weight is not a continuous image of the Tikhonov cube $ I ^ \tau $. Let $ f : X \rightarrow Y $ be a closed mapping of a space $ X $ onto a space $ T $. The mapping $ \mathop{\rm exp} f : \mathop{\rm exp} X \rightarrow \mathop{\rm exp} Y $ defined by $ ( \mathop{\rm exp} f ) ( \widehat{F} ) = ( f ( F) ) \widehat{ {}} $ is called the exponential mapping. If $ f : X \rightarrow Y $ is a continuous mapping of a compactum $ X $ onto a compactum $ Y $, then it is open if and only if the mapping $ \mathop{\rm exp} f $ is open. The functor $ \mathop{\rm exp} X $ acting from the category of compacta and continuous mappings into the same category is a covariant functor of exponential type. Here to a morphism $ f $ there corresponds its exponent $ \mathop{\rm exp} f $.

References

[1] K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) (Translated from French)

Comments

The exponential topology is better known as the Vietoris topology and the exponent of a space is usually called its hyperspace, cf. also Hyperspace. Concerning hyperspaces of Peano continua, it was shown in [a1] that these are in fact homeomorphic to the Hilbert cube.

References

[a1] D.W. Curtis, A.M. Schori, "Hyperspaces of Peano continua are Hilbert cubes" Fund. Math. , 101 (1978) pp. 19–38
How to Cite This Entry:
Exponential topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exponential_topology&oldid=46877
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article