Whitney decomposition

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)