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Proper pseudocompact extensions of compact abelian group topologies
Authors:
W. W. Comfort and Lewis C. Robertson
Journal:
Proc. Amer. Math. Soc. 86 (1982), 173-178
MSC:
Primary 22B05; Secondary 22D05, 54H99
MathSciNet review:
663891
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Abstract: A compact Abelian group admits a strictly finer pseudocompact topological group topology if and only if the weight of is uncountable.
- [1]
W. W. Comfort and Lewis C. Robertson, Cardinality constraints for pseudocompact and for totally dense subgroups of compact Abelian groups (in preparation).
- [2]
W.
W. Comfort and K.
A. Ross, Topologies induced by groups of characters, Fund.
Math. 55 (1964), 283–291. MR 0169940
(30 #183)
- [3]
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)
- [4]
W.
W. Comfort and T.
Soundararajan, Pseudocompact group topologies and totally dense
subgroups, Pacific J. Math. 100 (1982), no. 1,
61–84. MR
661441 (83m:22008)
- [5]
Leonard
Gillman and Meyer
Jerison, Rings of continuous functions, The University Series
in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton,
N.J.-Toronto-London-New York, 1960. MR 0116199
(22 #6994)
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Douglass
L. Grant, Topological groups which satisfy an open mapping
theorem, Pacific J. Math. 68 (1977), no. 2,
411–423. MR 0466395
(57 #6275)
- [7]
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis I, Grundlehren der math. Wissenschaften, vol. 115, Springer-Verlag, Berlin and New York, 1963.
- [8]
Edwin
Hewitt and Kenneth
A. Ross, Extensions of Haar measure and of harmonic analysis for
locally compact Abelian groups, Math. Ann. 160
(1965), 171–194. MR 0186751
(32 #4208)
- [9]
-, Abstract harmonic analysis II, Grundlehren der math. Wissenschaften, vol. 152, Springer-Verlag, Berlin and New York, 1970.
- [10]
M.
Rajagopalan, Topologies in locally compact groups, Math. Ann.
176 (1968), 169–180. MR 0224736
(37 #335)
- [11]
Neil
W. Rickert, Locally compact topologies for
groups, Trans. Amer. Math. Soc. 126 (1967), 225–235. MR 0202911
(34 #2770), http://dx.doi.org/10.1090/S0002-9947-1967-0202911-4
- [12]
K.
A. Ross, Closed subgroups of locally compact Abelian groups,
Fund. Math. 56 (1964), 241–244. MR 0171878
(30 #2104)
- [13]
T.
Soundararajan, Galois theory for general extension fields, J.
Reine Angew. Math. 241 (1970), 49–63. MR 0262211
(41 #6821)
- [14]
-, Totally dense subgroups of topological groups, General Topology and Its Relations to Modern Analysis and Algebra, III, (Proc. Topological Conf., Kanpur, 1968), Academia, Prague, 1971, pp. 299-300.
- [15]
T.
Soundararajan, Cohomology of Galois extensions, J. Pure Appl.
Algebra 11 (1977/78), no. 1-3, 139–150. MR 0466092
(57 #5975)
- [16]
-, Pseudocompact subgroups of Galois groups (to appear).
- [17]
R.
M. Stephenson Jr., Pseudocompact spaces, Trans. Amer. Math. Soc. 134 (1968), 437–448. MR 0232349
(38 #674), http://dx.doi.org/10.1090/S0002-9947-1968-0232349-6
- [18]
R.
M. Stephenson Jr., Minimal topological groups, Math. Ann.
192 (1971), 193–195. MR 0286934
(44 #4141)
- [19]
André Weil, Sur les espaces à structure uniforme et sur la topologie générale, Publ. Math. Univ. Strasbourg, Hermann, Paris, 1937.
- [20]
Howard
J. Wilcox, Pseudocompact groups, Pacific J. Math.
19 (1966), 365–379. MR 0206139
(34 #5964)
- [21]
Howard
J. Wilcox, Dense subgroups of compact
groups, Proc. Amer. Math. Soc. 28 (1971), 578–580. MR 0280640
(43 #6359), http://dx.doi.org/10.1090/S0002-9939-1971-0280640-5
- [1]
- W. W. Comfort and Lewis C. Robertson, Cardinality constraints for pseudocompact and for totally dense subgroups of compact Abelian groups (in preparation).
- [2]
- W. W. Comfort and K. A. Ross, Topologies induced by groups of characters, Fund. Math. 55 (1964), 283-291. MR 0169940 (30:183)
- [3]
- -, Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496. MR 0207886 (34:7699)
- [4]
- W. W. Comfort and T. Soundararajan, Pseudocompact group topologies and totally dense subgroups, Pacific J. Math. 100 (1982). MR 661441 (83m:22008)
- [5]
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, Van Nostrand, Princeton, N. J., 1960. MR 0116199 (22:6994)
- [6]
- Douglass L. Grant, Topological groups which satisfy an open mapping theorem, Pacific J. Math. 68 (1977), 411-423. MR 0466395 (57:6275)
- [7]
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis I, Grundlehren der math. Wissenschaften, vol. 115, Springer-Verlag, Berlin and New York, 1963.
- [8]
- -, Extensions of Haar measure and of harmonic analysis for locally compact Abelian groups, Math. Ann. 160 (1965), 171-194. MR 0186751 (32:4208)
- [9]
- -, Abstract harmonic analysis II, Grundlehren der math. Wissenschaften, vol. 152, Springer-Verlag, Berlin and New York, 1970.
- [10]
- M. Rajagopalan, Topologies in locally compact groups, Math. Ann. 176 (1968), 169-180. MR 0224736 (37:335)
- [11]
- N. W. Rickert, Locally compact topologies for groups, Trans. Amer. Math. Soc. 126 (1967), 225-235. MR 0202911 (34:2770)
- [12]
- K. A. Ross, Closed subgroups of locally compact Abelian groups, Fund. Math. 56 (1965), 241-244. MR 0171878 (30:2104)
- [13]
- T. Soundararajan, Galois theory for general extension fields, Crelle J. 241 (1970), 49-63. MR 0262211 (41:6821)
- [14]
- -, Totally dense subgroups of topological groups, General Topology and Its Relations to Modern Analysis and Algebra, III, (Proc. Topological Conf., Kanpur, 1968), Academia, Prague, 1971, pp. 299-300.
- [15]
- -, Cohomology of Galois extensions, J. Pure Appl. Algebra 11 (1977), 139-150. MR 0466092 (57:5975)
- [16]
- -, Pseudocompact subgroups of Galois groups (to appear).
- [17]
- R. M. Stephenson, Jr., Pseudocompact spaces, Trans. Amer. Math. Soc. 134 (1968), 437-448. MR 0232349 (38:674)
- [18]
- -, Minimal topological groups, Math. Ann. 192 (1971), 193-195. MR 0286934 (44:4141)
- [19]
- André Weil, Sur les espaces à structure uniforme et sur la topologie générale, Publ. Math. Univ. Strasbourg, Hermann, Paris, 1937.
- [20]
- Howard J. Wilcox, Pseudocompact groups, Pacific J. Math. 19 (1966), 365-379. MR 0206139 (34:5964)
- [21]
- -, Dense subgroups of compact groups, Proc. Amer. Math. Soc. 28 (1971), 578-580. MR 0280640 (43:6359)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1982-0663891-4
PII:
S 0002-9939(1982)0663891-4
Keywords:
Compact Abelian group,
pseudocompact group
Article copyright:
© Copyright 1982 American Mathematical Society
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