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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 AR-property for Roberts' example
of a compact convex set with no extreme points
Part 1: general result


Authors: Nguyen To Nhu, Jose M. R. Sanjurjo and Tran Van An
Journal: Proc. Amer. Math. Soc. 125 (1997), 3075-3087
MSC (1991): Primary 54C55; Secondary 54D45
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Abstract: We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hilbert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this to show that the example of Roberts is an AR.


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Additional Information

Nguyen To Nhu
Affiliation: Institute of Mathematics, P.O. Box 631, Bo Ho, Hanoi, Vietnam
Address at time of publication: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001
Email: nnguyen@nmsu.edu

Jose M. R. Sanjurjo
Affiliation: Departamento de Geometria y Topologia, Facultad de Matematicas, Universidad Complutense de Madrid, 280 40 Madrid, Spain
Email: sanjurjo@sungt1.mat.ucm.es

Tran Van An
Affiliation: Department of Mathematics, University of Vinh, Nghe An, Vietnam

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04020-3
PII: S 0002-9939(97)04020-3
Keywords: Convex set, linear metric space, extreme point, absolute retract
Received by editor(s): December 17, 1992
Received by editor(s) in revised form: April 1, 1996
Additional Notes: The first author was supported by the Complutense University of Madrid.
Communicated by: James West
Article copyright: © Copyright 1997 American Mathematical Society