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

DOI:
https://doi.org/10.1090/S0002-9939-97-04020-3

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Abstract | References | Similar Articles | Additional Information

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.

**[BD]**C. Bessaga and T. Dobrowolski, Some open problems on the border of infinite dimensional topology and functional analysis,*Proceedings of the international conference on geometric topology*, PWN, Warszawa 1980.**[CDM]**D. Curtis, T. Dobrowolski and J. Mogilski, Some applications of the topological characterizations of the sigma-compact spaces nad ,*Trans. Amer. Math. Soc.***284**(1984), 837-847. MR**86i:54035****[DM]**T. Dobrowolski and J. Mogilski, Problems on topological classification of incomplete metric spaces, Open problems in topology, J. van Mill and G. M. Reed (Editors) Elsevier Science Publishers B. V. North-Holland 1990. MR**92c:54001****[G]**R. Geoghegan, Open problems in infinite dimensional topology,*Topology Proceedings*,**4**(1979), 287-330. MR**82a:57015****[K1]**V. Klee, Shrinkable neighbourhoods in Hausdorff linear spaces,*Math. Ann.***141**(1960), 281-285. MR**24:A1003****[K2]**V. Klee, Leray-Schauder theory without local convexity,*Math. Ann.***141**(1960), 286-296. MR**24:A1004****[KM]**M. G. Krein and D. P. Milman, On extreme points of regular convex sets,*Studia Math.***9**(1940), 133-138. MR**3:90a****[KP]**N. J. Kalton and N. T. Peck, A re-examination of Roberts' example of a compact convex set with no extreme points,*Math. Ann.***253**(1980), 89-101. MR**82h:46055****[KPR]**N. J. Kalton, N. T. Peck and J. W. Roberts, An -space sampler,*London Math. Soc. Lecture Note Series*, vol. 89 Cambridge Univ. Press, 1984. MR**87c:46002****[N1]**Nguyen To Nhu, Investigating the ANR-property of metric spaces,*Fund. Math.***124**(1984), 243-254; Correction,*Fund. Math.***141**(1992), 297. MR**86d:54018**; MR**93k:54042****[N2]**Nguyen To Nhu, The finite dimensional approximation property and the AR-property in needle point spaces,*J. London Math. Soc.*(to appear).**[NS]**Nguyen To Nhu and Katsuro Sakai, The compact neighborhood extension property and the local equi-connectedness,*Proc. Amer. Math. Soc.***121**(1994), 259-265. MR**94g:54009****[NT1]**Nguyen To Nhu and Le Hoang Tri, Every needle point space contains a compact convex AR-set with no extreme points,*Proc. Amer. Math. Soc.***120**(1994), 1261-1265. MR**94f:54038****[NT2]**Nguyen To Nhu and Le Hoang Tri, No Roberts space is a counter-example to Schauder's conjecture,*Topology*,**33**(1994), 371-378. MR**95h:46014****[R1]**J. W. Roberts, A compact convex set with no extreme points,*Studia Math.***60**(1977), 255-266. MR**57:10595****[R2]**J. W. Roberts, Pathological compact convex sets in the spaces , The Altgeld Book, University of Illinois, 1976.**[Re]**S. Rolewicz, Metric linear spaces, PWN, Warszawa 1972; Second publication, PWN, Warszawa 1982. MR**55:10993**; MR**88i:46004a**; MR**88i:46004b**

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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:
https://doi.org/10.1090/S0002-9939-97-04020-3

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