# Whitney mapping

Let $X$ be a compact Hausdorff space. The hyperspace of $X$ is denoted by $2 ^ {X}$; the subspace of $2 ^ {X}$, consisting of all sub-continua of $X$ is denoted by $C ( X )$. A Whitney mapping of $X$ is a real-valued continuous function $w$ on $2 ^ {X}$( or on $C ( X )$) with the following properties:

1) $w ( \{ x \} ) = 0$ for each $x \in X$;

2) if $A$ and $B$ are in the domain of $w$ and if $A$ is a proper subset of $B$, then $w ( A ) < w ( B )$.

A set of type $w ^ {- 1 } ( t )$, for $0 \leq t < w ( tX )$, is called a Whitney level.

The existence of a Whitney function on $2 ^ {X}$ implies that $X$ is a $G _ \delta$- subset of its hyperspace, which in turn implies that $X$ is metrizable (cf. also Metrizable space). One of the simplest constructions of a Whitney mapping for a compact metrizable space $X$ is as follows. Let ${\mathcal O}$ be a countable open base of $X$, and, for each pair $U,V \in {\mathcal O}$ such that ${ \mathop{\rm Cl} } ( U ) \subseteq V$, fix a mapping $f : X \rightarrow {[0,1] }$ which equals $0$ on $U$ and $1$ outside $V$. Enumerate these functions as $( f _ {n} ) _ {n = 1 } ^ \infty$ and let

$$w ( A ) = \sum _ {n = 1 } ^ \infty 2 ^ {- n } { \mathop{\rm diam} } f _ {n} ( A ) .$$

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 $2 ^ {X}$, and led to a proof that $2 ^ {X}$ is acyclic and that both $C ( X )$ and $2 ^ {X}$ are even contractible if $X$ is Peanian (1942). Later on they became a subject of study in their own right.

A Whitney property is a topological property ${\mathcal P}$ such that if a metric continuum $X$ has ${\mathcal P}$, then so does each Whitney level of it in $C ( X )$. 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://encyclopediaofmath.org/index.php?title=Whitney_mapping&oldid=49212
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