Whitney mapping
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
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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 |
Whitney mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whitney_mapping&oldid=42892