Varieties generated by countably compact Abelian groups
Authors:
Dikran Dikranjan and Michael Tkachenko
Journal:
Proc. Amer. Math. Soc. 130 (2002), 24872496
MSC (1991):
Primary 22A05, 22B05, 54D25, 54H11; Secondary 54A35, 54B30, 54D30, 54H13
Published electronically:
February 4, 2002
MathSciNet review:
1897476
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We prove under the assumption of Martin's Axiom that every precompact Abelian group of size belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.
 [Ar1]
A.
V. Arhangel′skiĭ, Cardinal invariants of topological
groups. Embeddings and condensations, Dokl. Akad. Nauk SSSR
247 (1979), no. 4, 779–782 (Russian). MR 553825
(80m:54005)
 [Ar2]
A.
V. Arkhangel′skiĭ, Any topological group is a quotient
group of a zerodimensional topological group, Dokl. Akad. Nauk SSSR
258 (1981), no. 5, 1037–1040 (Russian). MR 631912
(83e:22002)
 [CM]
W.
W. Comfort and Jan
van Mill, On the existence of free topological groups,
Topology Appl. 29 (1988), no. 3, 245–265. MR 953957
(90e:22001), http://dx.doi.org/10.1016/01668641(88)900247
 [CR]
W.
W. Comfort and Kenneth
A. Ross, Pseudocompactness and uniform continuity in topological
groups, Pacific J. Math. 16 (1966), 483–496. MR 0207886
(34 #7699)
 [D]
Dikran
Dikranjan, Quotients of zerodimensional precompact abelian
groups, Topology Appl. 86 (1998), no. 1,
47–62. Special issue on topological groups. MR 1619342
(99e:54028), http://dx.doi.org/10.1016/S01668641(97)001284
 [DPS]
Dikran
N. Dikranjan, Ivan
R. Prodanov, and Luchezar
N. Stoyanov, Topological groups, Monographs and Textbooks in
Pure and Applied Mathematics, vol. 130, Marcel Dekker, Inc., New York,
1990. Characters, dualities and minimal group topologies. MR 1015288
(91e:22001)
 [DT1]
Dikran
Dikranjan and Michael
Tkačenko, Sequentially complete groups: dimension and
minimality, J. Pure Appl. Algebra 157 (2001),
no. 23, 215–239. MR 1812053
(2001m:22002), http://dx.doi.org/10.1016/S00224049(00)000153
 [DT2]
D. Dikranjan and M.G. Tkachenko, Sequential completeness of quotient groups, Bull. Austral. Math. Soc. 61 (2000), 129151. CMP 2001:10
 [DT3]
D. Dikranjan and M.G. Tkachenko, Weakly complete free topological groups, Topology Appl. 112 (2001), no. 3, 259287. MR 2001:11
 [DT4]
D. Dikranjan and M.G. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math, to appear.
 [DTT]
D.
Dikranjan, M.
Tkačenko, and V.
Tkachuk, Topological groups with thin generating sets, J. Pure
Appl. Algebra 145 (2000), no. 2, 123–148. MR 1733248
(2000m:22003), http://dx.doi.org/10.1016/S00224049(98)000759
 [Do1]
Eric
K. van Douwen, The product of two countably compact
topological groups, Trans. Amer. Math. Soc.
262 (1980), no. 2,
417–427. MR
586725 (82b:22002), http://dx.doi.org/10.1090/S00029947198005867258
 [Do2]
Eric
K. van Douwen, The maximal totally bounded group topology on
𝐺 and the biggest minimal 𝐺space, for abelian groups
𝐺, Topology Appl. 34 (1990), no. 1,
69–91. MR
1035461 (91d:54044), http://dx.doi.org/10.1016/01668641(90)90090O
 [Flo]
Peter
Flor, Zur BohrKonvergenz von Folgen, Math. Scand.
23 (1968), 169–170 (1969) (German). MR 0251457
(40 #4685)
 [Fuc]
László
Fuchs, Infinite abelian groups. Vol. I, Pure and Applied
Mathematics, Vol. 36, Academic Press, New YorkLondon, 1970. MR 0255673
(41 #333)
 [HJ]
A.
Hajnal and I.
Juhász, A separable normal topological group need not be
Lindelöf, General Topology and Appl. 6 (1976),
no. 2, 199–205. MR 0431086
(55 #4088)
 [HR]
Edwin
Hewitt and Kenneth
A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed.,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences], vol. 115, SpringerVerlag, BerlinNew York,
1979. Structure of topological groups, integration theory, group
representations. MR 551496
(81k:43001)
 [HM]
Klaas
Pieter Hart and Jan
van Mill, A countably compact topological group
𝐻 such that 𝐻×𝐻 is not countably
compact, Trans. Amer. Math. Soc.
323 (1991), no. 2,
811–821. MR
982236 (91e:54025), http://dx.doi.org/10.1090/S00029947199109822363
 [Juh]
I.
Juhász, Cardinal functions in topology, Mathematisch
Centrum, Amsterdam, 1971. In collaboration with A. Verbeek and N. S.
Kroonenberg; Mathematical Centre Tracts, No. 34. MR 0340021
(49 #4778)
 [Kuz]
V.
Kuz′minov, Alexandrov’s hypothesis in the theory of
topological groups, Dokl. Akad. Nauk SSSR 125 (1959),
727–729 (Russian). MR 0104753
(21 #3506)
 [Mo1]
S.A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145160. MR 41:3655
 [Mo2]
Sidney
A. Morris, Varieties of topological groups: a survey, Colloq.
Math. 46 (1982), no. 2, 147–165. MR 678128
(84e:22005)
 [Sha]
Dmitrii
B. Shakhmatov, Imbeddings into topological groups preserving
dimensions, Topology Appl. 36 (1990), no. 2,
181–204. Seminar on General Topology and Topological Algebra (Moscow,
1988/1989). MR
1068169 (91i:54028), http://dx.doi.org/10.1016/01668641(90)90008P
 [Tay]
Walter
Taylor, Varieties of topological algebras, J. Austral. Math.
Soc. Ser. A 23 (1977), no. 2, 207–241. MR 0447077
(56 #5392)
 [Tk]
Mikhail
Tkačenko, Introduction to topological groups, Topology
Appl. 86 (1998), no. 3, 179–231. MR 1623960
(99b:54064), http://dx.doi.org/10.1016/S01668641(98)000510
 [TY]
M.G. Tkachenko and Iv. Yaschenko, Independent group topologies on Abelian groups, Topology Appl., to appear.
 [To1]
Artur
Hideyuki Tomita, On finite powers of countably compact groups,
Comment. Math. Univ. Carolin. 37 (1996), no. 3,
617–626. MR 1426926
(98a:54033)
 [To2]
Artur
Hideyuki Tomita, A group under
𝑀𝐴_{𝑐𝑜𝑢𝑛𝑡𝑎𝑏𝑙𝑒}
whose square is countably compact but whose cube is not, Topology
Appl. 91 (1999), no. 2, 91–104. MR 1664516
(2000d:54039), http://dx.doi.org/10.1016/S01668641(97)00206X
 [Ar1]
 A.V. Arhangel'skii, Cardinal invariants of topological groups. Embeddings and condensations, Soviet Math. Dokl. 20 (1979), 783787. Russian original in: Dokl. AN SSSR 247 (1979), 779782. MR 80m:54005
 [Ar2]
 A.V. Arhangel'skii, Every topological group is a quotient group of a zerodimensional topological group, Dokl. AN SSSR 258 (1981), 10371040 (in Russian). MR 83e:22002
 [CM]
 W.W. Comfort and J. van Mill, On the existence of free topological groups, Topology Appl. 29 (1988), 245265. MR 90e:22001
 [CR]
 W.W. Comfort and K.A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483496. MR 34:7699
 [D]
 D. Dikranjan, Quotients of zerodimensional precompact abelian groups, Topology Appl. 86 (1998), no. 1, 4762. MR 99e:54028
 [DPS]
 D. Dikranjan, I. Prodanov and L. Stoyanov, Topological groups (Characters, Dualities and Minimal group topologies). Marcel Dekker, Inc., New YorkBasel, 1990. MR 91e:22001
 [DT1]
 D. Dikranjan and M.G Tkachenko, Sequentially complete groups: dimension and minimality, J. Pure Appl. Algebra 157 (2001), 215239. MR 2001m:22002
 [DT2]
 D. Dikranjan and M.G. Tkachenko, Sequential completeness of quotient groups, Bull. Austral. Math. Soc. 61 (2000), 129151. CMP 2001:10
 [DT3]
 D. Dikranjan and M.G. Tkachenko, Weakly complete free topological groups, Topology Appl. 112 (2001), no. 3, 259287. MR 2001:11
 [DT4]
 D. Dikranjan and M.G. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math, to appear.
 [DTT]
 D. Dikranjan, M.G. Tkachenko and V.V. Tkachuk, Topological groups with thin generating sets, J. Pure Appl. Algebra, 145 (2000) 123148. MR 2000m:22003
 [Do1]
 E. van Douwen, The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262 (1980), 417427. MR 82b:22002
 [Do2]
 E. van Douwen, The maximal totally bounded group topology on and the biggest minimal space for Abelian groups , Topology Appl. 34 (1990), 6991. MR 91d:54044
 [Flo]
 P. Flor, Zur BohrKonvergenz von Folgen, Math. Scand. 23 (1968), 169170. MR 40:4685
 [Fuc]
 L. Fuchs, Infinite Abelian groups, Vol. I, Academic Press, New York, 1970. MR 41:333
 [HJ]
 A. Hajnal and I. Juhász, A normal separable group need not be Lindelöf, Gen. Topol. Appl. 6 (1976), 199205. MR 55:4088
 [HR]
 E. Hewitt and K. Ross, Abstract Harmonic Analysis I (SpringerVerlag, BerlinGöttingenHeidelberg 1979). MR 81k:43001
 [HM]
 K. Hart and J.van Mill, A countably compact group such that is not countably compact, Trans. Amer. Math. Soc. 323 (1991), 811821. MR 91e:54025
 [Juh]
 I. Juhász, Cardinal functions in Topology, Math. Centre Tracts 34, Amsterdam 1971. MR 49:4778
 [Kuz]
 V. Kuz'minov, On a hypothesis of P.S. Alexandrov in the theory of topological groups, Doklady Akad. Nauk SSSR 125 (1959), 727729 (in Russian). MR 21:3506
 [Mo1]
 S.A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145160. MR 41:3655
 [Mo2]
 S.A. Morris, Varieties of topological groups. A survey. Colloq. Math. 46 (1982), 147165. MR 84e:22005
 [Sha]
 D. Shakhmatov, Imbeddings into topological groups preserving dimensions, Topology Appl. 36 (1990), 181204. MR 91i:54028
 [Tay]
 W. Taylor, Varieties of topological algebras, J. Austral. Math. Soc. 23 (1977), 207241. MR 56:5392
 [Tk]
 M.G. Tkachenko, Introduction to topological groups, Topology Appl. 86 (1998), 179231. MR 99b:54064
 [TY]
 M.G. Tkachenko and Iv. Yaschenko, Independent group topologies on Abelian groups, Topology Appl., to appear.
 [To1]
 A.H. Tomita, On finite powers of countably compact groups, Comment. Math. Univ. Carolin. 37 (1996), no. 3, 617626. MR 98a:54033
 [To2]
 A.H. Tomita, A group under whose square is countably compact but whose cube is not, Topology Appl. 91 (1999), 91104. MR 2000d:54039
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Additional Information
Dikran Dikranjan
Affiliation:
Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy
Email:
dikranja@dimi.uniud.it
Michael Tkachenko
Affiliation:
Departamento de Matemáticas, Universidad Autónoma Metropolitana, México
Email:
mich@xanum.uam.mx
DOI:
http://dx.doi.org/10.1090/S0002993902063542
PII:
S 00029939(02)063542
Keywords:
Countably compact,
precompact,
sequentially complete,
variety of topological groups,
Martin's Axiom
Received by editor(s):
February 4, 2000
Received by editor(s) in revised form:
March 21, 2001
Published electronically:
February 4, 2002
Additional Notes:
The first author was partially supported by Research Grant of the Italian MURST in the framework of the project “Nuove prospettive nella teoria degli anelli, dei moduli e dei gruppi abeliani" 2000
The second author was partially supported by the Mexican National Council of Sciences and Technology (CONACyT), grant no. 40020053012PE. He also thanks the hosts for the hospitality and generous support during his visit to the Università di Udine, Italy in December, 1999
Communicated by:
Alan Dow
Article copyright:
© Copyright 2002
American Mathematical Society
