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Let  $  X $
be a compact [[Hausdorff space|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|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 [[#References|[a4]]] in 1933 in a different context. They were first used by J.L. Kelley [[#References|[a2]]] to study hyperspaces. Every metric continuum can occur as a Whitney level [[#References|[a1]]]. For an account of continua theory see [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.J. Charatonik,   "Continua as positive Whitney levels"  ''Proc. Amer. Math. Soc.'' , '''118'''  (1993)  pp. 1351–1352</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,   "Hyperspaces of a continuum"  ''Trans. Amer. Math. Soc.'' , '''52'''  (1942)  pp. 22–36</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.B. Nadler,   "Hyperspaces of sets" , M. Dekker  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Whitney,   "Regular families of curves"  ''Ann. of Math.'' , '''2'''  (1933)  pp. 244–270</TD></TR></table>