Continua with unique symmetric product

José G. Anaya; Enrique Castañeda-Alvarado; Alejandro Illanes

Commentationes Mathematicae Universitatis Carolinae (2013)

  • Volume: 54, Issue: 3, page 397-406
  • ISSN: 0010-2628

Abstract

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Let X be a metric continuum. Let F n ( X ) denote the hyperspace of nonempty subsets of X with at most n elements. We say that the continuum X has unique hyperspace F n ( X ) provided that the following implication holds: if Y is a continuum and F n ( X ) is homeomorphic to F n ( Y ) , then X is homeomorphic to Y . In this paper we prove the following results: (1) if X is an indecomposable continuum such that each nondegenerate proper subcontinuum of X is an arc, then X has unique hyperspace F 2 ( X ) , and (2) let X be an arcwise connected continuum for which there exists a unique point v ∈ X such that v is the vertex of a simple triod. Then X has unique hyperspace F 2 ( X ) .

How to cite

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Anaya, José G., Castañeda-Alvarado, Enrique, and Illanes, Alejandro. "Continua with unique symmetric product." Commentationes Mathematicae Universitatis Carolinae 54.3 (2013): 397-406. <http://eudml.org/doc/260680>.

@article{Anaya2013,
abstract = {Let $X$ be a metric continuum. Let $F_\{n\}(X)$ denote the hyperspace of nonempty subsets of $X$ with at most $n$ elements. We say that the continuum $X$ has unique hyperspace $F_\{n\}(X)$ provided that the following implication holds: if $Y$ is a continuum and $F_\{n\}(X)$ is homeomorphic to $F_\{n\}(Y)$, then $X$ is homeomorphic to $Y$. In this paper we prove the following results: (1) if $X$ is an indecomposable continuum such that each nondegenerate proper subcontinuum of $X$ is an arc, then $X$ has unique hyperspace $F_\{2\}(X)$, and (2) let $X$ be an arcwise connected continuum for which there exists a unique point $v\in X$ such that $v$ is the vertex of a simple triod. Then $X$ has unique hyperspace $F_\{2\}(X)$.},
author = {Anaya, José G., Castañeda-Alvarado, Enrique, Illanes, Alejandro},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {arc continuum; continuum; indecomposable; symmetric product; unique hyperspace; arc continuum; indecomposable; symmetric product; unique hyperspace},
language = {eng},
number = {3},
pages = {397-406},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Continua with unique symmetric product},
url = {http://eudml.org/doc/260680},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Anaya, José G.
AU - Castañeda-Alvarado, Enrique
AU - Illanes, Alejandro
TI - Continua with unique symmetric product
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 3
SP - 397
EP - 406
AB - Let $X$ be a metric continuum. Let $F_{n}(X)$ denote the hyperspace of nonempty subsets of $X$ with at most $n$ elements. We say that the continuum $X$ has unique hyperspace $F_{n}(X)$ provided that the following implication holds: if $Y$ is a continuum and $F_{n}(X)$ is homeomorphic to $F_{n}(Y)$, then $X$ is homeomorphic to $Y$. In this paper we prove the following results: (1) if $X$ is an indecomposable continuum such that each nondegenerate proper subcontinuum of $X$ is an arc, then $X$ has unique hyperspace $F_{2}(X)$, and (2) let $X$ be an arcwise connected continuum for which there exists a unique point $v\in X$ such that $v$ is the vertex of a simple triod. Then $X$ has unique hyperspace $F_{2}(X)$.
LA - eng
KW - arc continuum; continuum; indecomposable; symmetric product; unique hyperspace; arc continuum; indecomposable; symmetric product; unique hyperspace
UR - http://eudml.org/doc/260680
ER -

References

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  9. Illanes A., Dendrites with unique hyperspace F 2 ( X ) , JP J. Geom. Topol. 2 (2002), 75–96. Zbl1025.54021MR1942627
  10. Illanes A., Uniqueness of hyperspaces, Questions Answers Gen. Topology 30 (2012), 21–44. MR2954279
  11. Illanes A., Models of hyperspaces, Topology Proc. 41 (2013), 39–64. MR2920967
  12. Illanes A., Martínez-Montejano J.M., Compactifications of [ 0 , ∞ ) with unique hyperspace F n ( X ) , Glas. Mat. Ser. III 44 (64) (2009), 457–478. Zbl1185.54008MR2587312
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