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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Space of subcontinua of a 2-dimensional continuum is infinite dimensional


Authors: Michael Levin and Yaki Sternfeld
Journal: Proc. Amer. Math. Soc. 125 (1997), 2771-2775
MSC (1991): Primary 54B20, 54F15, 54F45
MathSciNet review: 1425131
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Abstract: Let $X$ be a metric continuum and let ${\mathcal C}(X)$ denote the space of subcontinua of $X$ with the Hausdorff metric. We settle a longstanding problem showing that if $\dim X = 2$ then $\dim {\mathcal C}(X)= \infty $. The special structure and properties of hereditarily indecomposable continua are applied in the proof.


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Additional Information

Michael Levin
Affiliation: Department of Mathematics, Haifa University, Mount Carmel, Haifa 31905, Israel
Email: levin@mathcs2.haifa.ac.il

Yaki Sternfeld
Affiliation: Department of Mathematics, Haifa University, Mount Carmel, Haifa 31905, Israel
Email: yaki@mathcs2.haifa.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04172-5
PII: S 0002-9939(97)04172-5
Keywords: Hyperspaces, hereditarily indecomposable continua, $2$-dimensional continua
Received by editor(s): February 4, 1995
Received by editor(s) in revised form: March 31, 1996
Communicated by: James West
Article copyright: © Copyright 1997 American Mathematical Society