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The hyperspace C2(X), for a finite graph X, is unique
by
Alejandro Illanes
Universidad Nacional Autónoma de México
A continuum is a compact connected metric space. For a continuum X, let Cn(X) denote the hyperspace of nonempty closed subsets of X which contain at most n components, the hyperspace Cn(X) is considered with the Hausdorff metric. From the work by Duda in 1968, it is known that if X is a finite graph different from an arc and a simple closed curve and Y is a continuum such that C1(X) is homeomorphic to C1(Y), then X is homeomorphic to Y. In this paper we show the following result:
Theorem If X is a finite graph and Y is a continuum such that C2(X) is homeomorphic to C2(Y), then X is homeomorphic to Y.
The respective question for n > 2 is still open. We will present some other open questions.
Date received: January 31, 2000
Copyright © 2000 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 # cady-22.