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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100401.png" /> be a compact [[Hausdorff space|Hausdorff space]]. The hyperspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100402.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100403.png" />; the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100404.png" />, consisting of all sub-continua of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100405.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100406.png" />. A Whitney mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100407.png" /> is a real-valued continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100408.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100409.png" /> (or on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w11004010.png" />) with the following properties:
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100401.png" /> be a compact [[Hausdorff space|Hausdorff space]]. The [[hyperspace]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100402.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100403.png" />; the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100404.png" />, consisting of all sub-continua of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100405.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100406.png" />. A Whitney mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100407.png" /> is a real-valued continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100408.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w1100409.png" /> (or on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w11004010.png" />) with the following properties:
  
 
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w11004011.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w11004012.png" />;
 
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w11004011.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110040/w11004012.png" />;

Latest revision as of 23:19, 2 March 2018

Let be a compact Hausdorff space. The hyperspace of is denoted by ; the subspace of , consisting of all sub-continua of is denoted by . A Whitney mapping of is a real-valued continuous function on (or on ) with the following properties:

1) for each ;

2) if and are in the domain of and if is a proper subset of , then .

A set of type , for , is called a Whitney level.

The existence of a Whitney function on implies that is a -subset of its hyperspace, which in turn implies that is metrizable (cf. also Metrizable space). One of the simplest constructions of a Whitney mapping for a compact metrizable space is as follows. Let be a countable open base of , and, for each pair such that , fix a mapping which equals on and outside . Enumerate these functions as and let

Below it is assumed that all spaces are metric continua. Whitney functions have been developed as a fundamental tool in continua theory. Their first use in this context was involved with order arcs in , and led to a proof that is acyclic and that both and are even contractible if is Peanian (1942). Later on they became a subject of study in their own right.

A Whitney property is a topological property such that if a metric continuum has , then so does each Whitney level of it in . Examples are: being a (locally connected) continuum; being a hereditarily indecomposable continuum; being a (pseudo-) arc or a circle; etc. Counterexamples are: being contractible; being acyclic in Alexander–Čech cohomology; being homogeneous; being a Hilbert cube.

Whitney functions were introduced by H. Whitney [a4] in 1933 in a different context. They were first used by J.L. Kelley [a2] to study hyperspaces. Every metric continuum can occur as a Whitney level [a1]. For an account of continua theory see [a3].

References

[a1] W.J. Charatonik, "Continua as positive Whitney levels" Proc. Amer. Math. Soc. , 118 (1993) pp. 1351–1352
[a2] J.L. Kelley, "Hyperspaces of a continuum" Trans. Amer. Math. Soc. , 52 (1942) pp. 22–36
[a3] S.B. Nadler, "Hyperspaces of sets" , M. Dekker (1978)
[a4] H. Whitney, "Regular families of curves" Ann. of Math. , 2 (1933) pp. 244–270
How to Cite This Entry:
Whitney mapping. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Whitney_mapping&oldid=13063
This article was adapted from an original article by M. van de Vel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article