Whitney decomposition

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

, the set of subsets that are closed and non-empty; and

, the set of subsets that are connected. Both sets are considered with the Hausdorff metric.

A Whitney mapping for is a continuous function from to the closed unit interval such that , for each point and, if and is a proper subset of , then .

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 .

A Whitney level is a fibre of the restriction to of a Whitney mapping for , that is, Whitney levels are sets of the form , where is a Whitney mapping for and .

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

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

Furthermore, given a fixed Whitney mapping , the set is a very nice (continuous) decomposition of the hyperspace . 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) ; then it is possible to consider the space of Whitney decompositions, . In [a1] it was proved that for every continuum , is homeomorphic to the Hilbert linear space .

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)