Volume 34,2009
Pages 175{190
http://topology.auburn.edu/tp/
Dendrites with Unique
Symmetric Products
by
David HerreraCarrasco,Mar
¶
³a de J.L
¶
opez,and
Fernando Mac
¶
³asRomero
Electronically published on June 11,2009
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° by Topology Proceedings.All rights reserved.
Volume 34 (2009)
Pages 175190
http://topology.auburn.edu/tp/
EPublished on June 11,2009
DENDRITES WITH UNIQUE
SYMMETRIC PRODUCTS
DAVID HERRERACARRASCO,MAR
¶
IA de J.L
¶
OPEZ,
AND FERNANDO MAC
¶
IASROMERO
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 nth symmetric
product of X and the nfold 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
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