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A continuum is a non-empty compact connected [[Metric space|metric space]]. A hyperspace of a continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201501.png" /> is a space whose elements are in a certain class of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201502.png" />. The most common hyperspaces are:
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A continuum is a non-empty compact connected [[Metric space|metric space]]. A hyperspace of a continuum $X$ is a space whose elements are in a certain class of subsets of $X$. The most common hyperspaces are:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201503.png" />, the set of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201504.png" /> that are closed and non-empty; and
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$2^X$, the set of subsets $A\subset X$ that are closed and non-empty; and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201505.png" />, the set of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201506.png" /> that are connected. Both sets are considered with the [[Hausdorff metric|Hausdorff metric]].
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$C(X)$, the set of subsets $A\in2^X$ that are connected. Both sets are considered with the [[Hausdorff metric|Hausdorff metric]].
  
A Whitney mapping for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201507.png" /> is a [[Continuous function|continuous function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201508.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w1201509.png" /> to the closed unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015012.png" /> for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015013.png" /> and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015015.png" /> is a proper subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015017.png" />.
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A Whitney mapping for $2^X$ is a [[Continuous function|continuous function]] $\mu$ from $2^X$ to the closed unit interval $[0,1]$ such that $\mu(X)=1$, $\mu(\{x\})=0$ for each point $x\in X$ and, if $A,B\in2^X$ and $A$ is a proper subset of $B$, then $\mu(A)<\mu(B)$.
  
Every continuum admits Whitney mappings [[#References|[a2]]], Thm. 13.4. These mappings are an important tool in the study of hyperspaces and they represent a way to give a  "size"  to the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015018.png" />.
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Every continuum admits Whitney mappings [[#References|[a2]]], Thm. 13.4. These mappings are an important tool in the study of hyperspaces and they represent a way to give a  "size"  to the elements of $2^X$.
  
A Whitney level is a fibre of the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015019.png" /> of a Whitney mapping for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015020.png" />, that is, Whitney levels are sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015022.png" /> is a Whitney mapping for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015024.png" />.
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A Whitney level is a fibre of the restriction to $C(X)$ of a Whitney mapping for $2^X$, that is, Whitney levels are sets of the form $\mu^{-1}(t)\cap C(X)$, where $\mu$ is a Whitney mapping for $2^X$ and $0<t<1$.
  
It is possible to consider the notion of Whitney level for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015025.png" />; these have not been very interesting, mainly because they are not necessarily connected [[#References|[a2]]], Thm. 24.2.
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It is possible to consider the notion of Whitney level for $2^X$; these have not been very interesting, mainly because they are not necessarily connected [[#References|[a2]]], Thm. 24.2.
  
In the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015026.png" />, Whitney levels are always compact and connected [[#References|[a2]]], Thm. 19.9, and they have many similarities with the continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015027.png" /> (see [[#References|[a2]]], Chap. VIII, for these similarities).
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In the case of $C(X)$, Whitney levels are always compact and connected [[#References|[a2]]], Thm. 19.9, and they have many similarities with the continuum $X$ (see [[#References|[a2]]], Chap. VIII, for these similarities).
  
Furthermore, given a fixed Whitney mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015028.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015029.png" /> is a very nice (continuous) decomposition of the hyperspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015030.png" />. A set of this form is called a Whitney decomposition.
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Furthermore, given a fixed Whitney mapping $\mu$, the set $\{\mu^{-1}(t)\cap C(X):t\in[0,1]\}$ is a very nice (continuous) decomposition of the hyperspace $C(X)$. A set of this form is called a Whitney decomposition.
  
A Whitney decomposition can be considered as an element of the hyperspace (of second order) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015031.png" />; then it is possible to consider the space of Whitney decompositions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015032.png" />. In [[#References|[a1]]] it was proved that for every continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015034.png" /> is homeomorphic to the Hilbert linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120150/w12015035.png" />.
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A Whitney decomposition can be considered as an element of the hyperspace (of second order) $C(C(X))$; then it is possible to consider the space of Whitney decompositions, $\operatorname{WD}(X)$. In [[#References|[a1]]] it was proved that for every continuum $X$, $\operatorname{WD}(X)$ is homeomorphic to the Hilbert linear space $l_2$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Illanes,  "The space of Whitney decompositions"  ''Ann. Inst. Mat. Univ. Nac. Autónoma México'' , '''28'''  (1988)  pp. 47–61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Illanes,  S.B. Nadler Jr.,  "Hyperspaces, fundamentals and recent advances" , ''Monogr. Textbooks Pure Appl. Math.'' , '''216''' , M. Dekker  (1999)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Illanes,  "The space of Whitney decompositions"  ''Ann. Inst. Mat. Univ. Nac. Autónoma México'' , '''28'''  (1988)  pp. 47–61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Illanes,  S.B. Nadler Jr.,  "Hyperspaces, fundamentals and recent advances" , ''Monogr. Textbooks Pure Appl. Math.'' , '''216''' , M. Dekker  (1999)</TD></TR></table>

Latest revision as of 15:02, 29 December 2018

A continuum is a non-empty compact connected metric space. A hyperspace of a continuum $X$ is a space whose elements are in a certain class of subsets of $X$. The most common hyperspaces are:

$2^X$, the set of subsets $A\subset X$ that are closed and non-empty; and

$C(X)$, the set of subsets $A\in2^X$ that are connected. Both sets are considered with the Hausdorff metric.

A Whitney mapping for $2^X$ is a continuous function $\mu$ from $2^X$ to the closed unit interval $[0,1]$ such that $\mu(X)=1$, $\mu(\{x\})=0$ for each point $x\in X$ and, if $A,B\in2^X$ and $A$ is a proper subset of $B$, then $\mu(A)<\mu(B)$.

Every continuum admits Whitney mappings [a2], Thm. 13.4. These mappings are an important tool in the study of hyperspaces and they represent a way to give a "size" to the elements of $2^X$.

A Whitney level is a fibre of the restriction to $C(X)$ of a Whitney mapping for $2^X$, that is, Whitney levels are sets of the form $\mu^{-1}(t)\cap C(X)$, where $\mu$ is a Whitney mapping for $2^X$ and $0<t<1$.

It is possible to consider the notion of Whitney level for $2^X$; these have not been very interesting, mainly because they are not necessarily connected [a2], Thm. 24.2.

In the case of $C(X)$, Whitney levels are always compact and connected [a2], Thm. 19.9, and they have many similarities with the continuum $X$ (see [a2], Chap. VIII, for these similarities).

Furthermore, given a fixed Whitney mapping $\mu$, the set $\{\mu^{-1}(t)\cap C(X):t\in[0,1]\}$ is a very nice (continuous) decomposition of the hyperspace $C(X)$. A set of this form is called a Whitney decomposition.

A Whitney decomposition can be considered as an element of the hyperspace (of second order) $C(C(X))$; then it is possible to consider the space of Whitney decompositions, $\operatorname{WD}(X)$. In [a1] it was proved that for every continuum $X$, $\operatorname{WD}(X)$ is homeomorphic to the Hilbert linear space $l_2$.

References

[a1] A. Illanes, "The space of Whitney decompositions" Ann. Inst. Mat. Univ. Nac. Autónoma México , 28 (1988) pp. 47–61
[a2] A. Illanes, S.B. Nadler Jr., "Hyperspaces, fundamentals and recent advances" , Monogr. Textbooks Pure Appl. Math. , 216 , M. Dekker (1999)
How to Cite This Entry:
Whitney decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitney_decomposition&oldid=12899
This article was adapted from an original article by A. Illanes (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article