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