The ARproperty 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), 30753087
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.
 [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]
Doug
Curtis, Tadeusz
Dobrowolski, and Jerzy
Mogilski, Some applications of the topological
characterizations of the sigmacompact spaces 𝑙²_{𝑓}
and Σ, Trans. Amer. Math. Soc.
284 (1984), no. 2,
837–846. MR
743748 (86i:54035), http://dx.doi.org/10.1090/S00029947198407437487
 [DM]
Jan
van Mill and George
M. Reed (eds.), Open problems in topology, NorthHolland
Publishing Co., Amsterdam, 1990. MR 1078636
(92c:54001)
 [G]
Open problems in infinitedimensional topology, The Proceedings
of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), 1979,
pp. 287–338 (1980). MR 583711
(82a:57015)
 [K1]
Victor
Klee, Shrinkable neighborhoods in Hausdorff linear spaces,
Math. Ann. 141 (1960), 281–285. MR 0131149
(24 #A1003)
 [K2]
Victor
Klee, LeraySchauder theory without local convexity, Math.
Ann. 141 (1960), 286–296. MR 0131150
(24 #A1004)
 [KM]
M.
Krein and D.
Milman, On extreme points of regular convex sets, Studia Math.
9 (1940), 133–138 (English., with Ukrainian
summary). MR
0004990 (3,90a)
 [KP]
N.
J. Kalton and N.
T. Peck, A reexamination of the Roberts example of a compact
convex set without extreme points, Math. Ann. 253
(1980), no. 2, 89–101. MR 597819
(82h:46055), http://dx.doi.org/10.1007/BF01578905
 [KPR]
N.
J. Kalton, N.
T. Peck, and James
W. Roberts, An 𝐹space sampler, London Mathematical
Society Lecture Note Series, vol. 89, Cambridge University Press,
Cambridge, 1984. MR 808777
(87c:46002)
 [N1]
Nguyen
To Nhu, Investigating the ANRproperty of metric spaces, Fund.
Math. 124 (1984), no. 3, 243–254. MR 774515
(86d:54018)
Nguyen
To Nhu, Corrections to: “Investigating the ANRproperty of
metric spaces” [Fund.\ Math.\ {124} (1984), no.\ 3, 243–254;
MR0774515 (86d:54018)], Fund. Math. 141 (1992),
no. 3, 297. MR 1199242
(93k:54042)
 [N2]
Nguyen To Nhu, The finite dimensional approximation property and the ARproperty in needle point spaces, J. London Math. Soc. (to appear).
 [NS]
Nguyen
To Nhu and Katsuro
Sakai, The compact neighborhood extension
property and local equiconnectedness, Proc.
Amer. Math. Soc. 121 (1994), no. 1, 259–265. MR 1232141
(94g:54009), http://dx.doi.org/10.1090/S00029939199412321410
 [NT1]
Nguyen
To Nhu and Le
Hoang Tri, Every needle point space contains a
compact convex ARset with no extreme points, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1261–1265. MR 1152989
(94f:54038), http://dx.doi.org/10.1090/S00029939199411529890
 [NT2]
Nguyen
To Nhu and Le
Hoang Tri, No Roberts space is a counterexample to Schauder’s
conjecture, Topology 33 (1994), no. 2,
371–378. MR 1273789
(95h:46014), http://dx.doi.org/10.1016/00409383(94)900183
 [R1]
James
W. Roberts, A compact convex set with no extreme points,
Studia Math. 60 (1977), no. 3, 255–266. MR 0470851
(57 #10595)
 [R2]
J. W. Roberts, Pathological compact convex sets in the spaces , The Altgeld Book, University of Illinois, 1976.
 [Re]
Stefan
Rolewicz, Metric linear spaces, PWNPolish Scientific
Publishers, Warsaw, 1972. Monografie Matematyczne, Tom. 56. [Mathematical
Monographs, Vol. 56]. MR 0438074
(55 #10993)
Stefan
Rolewicz, Metric linear spaces, 2nd ed., PWN—Polish
Scientific Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1984.
MR 802450
(88i:46004a)
Stefan
Rolewicz, Metric linear spaces, 2nd ed., Mathematics and its
Applications (East European Series), vol. 20, D. Reidel Publishing
Co., Dordrecht; PWN—Polish Scientific Publishers, Warsaw, 1985. MR 808176
(88i:46004b)
 [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 sigmacompact spaces nad , Trans. Amer. Math. Soc. 284(1984), 837847. 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. NorthHolland 1990. MR 92c:54001
 [G]
 R. Geoghegan, Open problems in infinite dimensional topology, Topology Proceedings, 4(1979), 287330. MR 82a:57015
 [K1]
 V. Klee, Shrinkable neighbourhoods in Hausdorff linear spaces, Math. Ann. 141(1960), 281285. MR 24:A1003
 [K2]
 V. Klee, LeraySchauder theory without local convexity, Math. Ann. 141(1960), 286296. MR 24:A1004
 [KM]
 M. G. Krein and D. P. Milman, On extreme points of regular convex sets, Studia Math. 9(1940), 133138. MR 3:90a
 [KP]
 N. J. Kalton and N. T. Peck, A reexamination of Roberts' example of a compact convex set with no extreme points, Math. Ann. 253(1980), 89101. 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 ANRproperty of metric spaces, Fund. Math. 124(1984), 243254; Correction, Fund. Math. 141(1992), 297. MR 86d:54018; MR 93k:54042
 [N2]
 Nguyen To Nhu, The finite dimensional approximation property and the ARproperty 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 equiconnectedness, Proc. Amer. Math. Soc. 121(1994), 259265. MR 94g:54009
 [NT1]
 Nguyen To Nhu and Le Hoang Tri, Every needle point space contains a compact convex ARset with no extreme points, Proc. Amer. Math. Soc. 120(1994), 12611265. MR 94f:54038
 [NT2]
 Nguyen To Nhu and Le Hoang Tri, No Roberts space is a counterexample to Schauder's conjecture, Topology, 33(1994), 371378. MR 95h:46014
 [R1]
 J. W. Roberts, A compact convex set with no extreme points, Studia Math. 60(1977), 255266. 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 880038001
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/S0002993997040203
PII:
S 00029939(97)040203
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
