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The AR-property for Roberts' example of a compact convex set with no extreme points .1. General result


Nhu, Nguyen Tho and Rodríguez Sanjurjo, José Manuel and Van An, Tran (1997) The AR-property for Roberts' example of a compact convex set with no extreme points .1. General result. Proceedings of the American Mathematical Society, 125 (10). pp. 3075-3087. ISSN 0002-9939

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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.

Item Type:Article
Uncontrolled Keywords:Convex set, linear metric space, extreme point, absolute retract
Subjects:Sciences > Mathematics > Geometry
Sciences > Mathematics > Topology
ID Code:16901

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.

D. Curtis, T. Dobrowolski and J. Mogilski, Some applications of the topological characterizations of the sigma-compact spaces 2 f nad , Trans. Amer. Math. Soc. 284(1984), 837{847. MR 86i:54035

T. Dobrowolski and J. Mogilski, Problems on topological classication of incomplete metric spaces, Open problems in topology, J. van Mill and G. M. Reed (Editors) Elsevier Science Publishers B. V. North-Holland 1990

R. Geoghegan, Open problems in infinite dimensional topology, Topology Proceedings, 4(1979), 287{330.

V. Klee, Shrinkable neighbourhoods in Hausdor linear spaces, Math. Ann. 141(1960), 281{285.

V. Klee, Leray-Schauder theory without local convexity, Math. Ann. 141(1960), 286{296.

M. G. Krein and D. P. Milman, On extreme points of regular convex sets, Studia Math. 9(1940), 133{138.

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.

N. J. Kalton, N. T. Peck and J. W. Roberts, An F-space sampler, London Math. Soc. Lecture Note Series, vol. 89 Cambridge Univ. Press, 1984.

Nguyen To Nhu, Investigating the ANR-property of metric spaces, Fund. Math. 124(1984), 243{254; Correction, Fund. Math. 141(1992), 297.

Nguyen To Nhu, The infinite dimensional approximation property and the AR-property in needle point spaces, J. London Math. Soc. (to appear).

Nguyen To Nhu and Katsuro Sakai, The compact neighborhood extension property and the local equi-connectedness, Proc. Amer. Math. Soc. 121(1994), 259{265.

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.

Nguyen To Nhu and Le Hoang Tri, No Roberts space is a counter-example to Schauder's conjecture, Topology, 33(1994), 371{378.

J. W. Roberts, A compact convex set with no extreme points, Studia Math. 60(1977), 255{266.

J. W. Roberts, Pathological compact convex sets in the spaces Lp; 0 p < 1, The Altgeld Book, University of Illinois, 1976.

S. Rolewicz, Metric linear spaces, PWN, Warszawa 1972; Second publication, PWN, Warszawa 1982.

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