Topology Proceedings
Document # baal-17
Topology Proceedings 26 No. 1 (2001-2002), pp. 213-233 |
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Size levels of hyperspaces
Alejandro Illanes
Let X be a metric continuum and C(X) the hyperspace of
subcontinua of X. A size map is a continuous function
\sigma:C(X) --> [0, \infty) such that \sigma({x})=0
for each x in X and, if A, B in C(X) and A subset B, then
\sigma(A) <= \sigma(B). A size level is a set of the form
\sigma-1(t), where \sigma is a size map and
t in [0, \sigma(X)]. It is known that size levels are subcontinua of
C(X), so we consider the space SL(X) of size levels as a subspace of
C(C(X)). In this paper we study the space SL(X) and obtain an
intrinsic characterization of size levels. As a consequence, we show that
SL([0, 1]) is not homeomorphic to the Hilbert space l2 and we obtain
topological characterizations of size levels of the hyperspaces of [0, 1]
and the unit circle in the plane.
Keywords: arc, continuum, hyperspace, simple closed curve, size level,
size map, Whitney level, Whitney map
Mathematics Subject Classification: 54B20, 54F20
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Topology Proceedings,
Volume 26 No. 1 (2001-2002)
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Date: November 1, 2002.