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II Congreso Iberoamericano de Topología y sus Aplicaciones
March 20-22, 1997

Morelía, Mexico

Organizers
Salvador García-Ferreira, Daniel Juan Pineda, Sergio Macías Alvarez, Max Neumann Coto, María L. Pérez Seguí, Salvador Romaguera Bonilla, Manuel Sanchis López, Angel Tamariz Mascarúa, M. G. Tkachenko, Javier F. Trigos Arrieta

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Chainable continua are not C-determined
by
Alejandro Illanes
Universidad Nacional Autónoma de México

A continuum is a compact connected metric space. For a continuum X, let C(X) denote the space of all subcontinua of X. The members of a class of continua \Lambda are said to be C-determined provided that if X and Y belong to \Lambda and C(X) \approx C(Y) (C(X) is homeomorphic to C(Y)), then X \approx Y. The members of the following classes are known to be C-determined:

  1. Finite graphs different from an arc (R. Duda),

  2. Hereditarily indecomposable continua (S. B. Nadler Jr.),

  3. Smooth fans (C. Eberhart and S. B. Nadler Jr.), and

  4. Indecomposable continua such that all their proper nondegenerate subcontinua are arcs (S. Macías).
Nadler has asked if the members of the class of chainable continua are C-determined. In this paper we answer this question in the negative and we construct an uncountable family of pairs of chainable continua F = { (XS, YS) | S in J } such that

  1. for each S in J, C(XS) \approx C(YS) and XS \not \approx YS, and

  2. if S =/= T, then XS \not \approx XT and YS \not \approx YT.

Date received: February 18, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caak-40.