Dendrites with Unique Symmetric Products

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Volume 34,2009
Pages 175{190
http://topology.auburn.edu/tp/
Dendrites with Unique
Symmetric Products
by
David Herrera-Carrasco,Mar

³a de J.L

opez,and
Fernando Mac

³as-Romero
Electronically published on June 11,2009
This ¯le contains only the ¯rst page of the paper.
The full version of the
paper is available to Topology Proceedings subscribers.
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Topology Proceedings
Web:http://topology.auburn.edu/tp/
Mail:Topology Proceedings
Department of Mathematics & Statistics
Auburn University,Alabama 36849,USA
E-mail:topolog@auburn.edu
ISSN:0146-4124
COPYRIGHT
c
° by Topology Proceedings.All rights reserved.
Volume 34 (2009)
Pages 175-190
http://topology.auburn.edu/tp/
E-Published on June 11,2009
DENDRITES WITH UNIQUE
SYMMETRIC PRODUCTS
DAVID HERRERA-CARRASCO,MAR

IA de J.L

OPEZ,
AND FERNANDO MAC

IAS-ROMERO
Abstract.
Let X be a metric continuum and n a positive
integer.Let F
n
(X) be the hyperspace of all nonempty subsets
of X with at most n points,metrized by the Hausdor® metric.
Let X be a dendrite whose set of end points is closed,let Y
be a continuum,and let n 2 N:In this paper,we prove that if
F
n
(X) is homeomorphic to F
n
(Y );then X is homeomorphic
to Y:
1.Introduction and general notions
A continuum is a nondegenerate,compact,connected metric
space.For a given continuum X and n 2 N;we consider the fol-
lowing hyperspaces of X:
F
n
(X) = fA ½ X:A is nonempty and has at most n pointsg
and
C
n
(X) = fA ½ X:A is closed nonempty
and has at most n componentsg:
Both F
n
(X) and C
n
(X) are metrized by the Hausdor® metric
(see [14,De¯nition 0.1]) and are also known as the n-th symmetric
product of X and the n-fold hyperspace of X,respectively.When
n = 1,it is customary to write C(X) instead of C
1
(X) and to refer
to C(X) as the hyperspace of subcontinua of X.
2000 Mathematics Subject Classi¯cation.54B20,54C50.
Key words and phrases.continuum,dendrite,unique hyperspace.
c
°2009 Topology Proceedings.
175
This ¯le contains only the ¯rst page of the paper.
The full version of the
paper is available to Topology Proceedings subscribers.
See
http://topology.auburn.edu/tp/subscriptioninfo.html
for information.