Abstract
We first study the extent of \({{\mathbb {Z}}}_b^{{\mathbb {R}}}\), the product of \({\mathfrak {c}}\)-copies of the group of integer numbers, each of them endowed with its Bohr topology. We prove that it is exactly \({\mathfrak {c}}\). From this fact we derive the non-normality of any locally quasi-convex group topology on \({{\mathbb {Z}}}^{{\mathbb {R}}}\) compatible with the duality \(({ {\mathbb {Z}}}^{{{\mathbb {R}}}}, {\mathbb {T}}^{({{\mathbb {R}}})})\). Finally we prove that any product of \({\mathfrak {c}}\)-many locally compact, non countably compact, separable abelian topological groups does not admit either a normal locally quasi-convex compatible topology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
23 June 2018
Unfortunately an erratum appears in the statement corresponding to Theorem 6.9 of the above mentioned paper.
References
Arhangel’skii, A.V., Tkachenko, M.G.: Topological Groups and Related Structures. Atlantis Press/World Scientific, Amsterdam/Paris (2008)
Außenhofer, L.: Contributions to the Duality Theory of Abelian Topological Groups and to the Theory of Nuclear Groups. Diss. Math. 384 (1999)
Außenhofer, L., Dikranjan, D., Martín-Peinador, E.: Locally quasi-convex compatible topologies on a topological group. Axioms 4(4), 436–458 (2015). https://doi.org/10.3390/axioms4040436
Außenhofer, L.: On the non-existence of the Mackey topology for locally quasi-convex groups. Forum Math. https://doi.org/10.1515/forum-2017-0179
Banaszczyk, W., Martín-Peinador, E.: Weakly pseudocompact subsets of nuclear groups. J. Pure Appl. Algebra 138, 99–106 (1999)
Chasco, M.J., Martín-Peinador, E., Tarieladze, V.: On Mackey topology for groups. Stud. Math. 132(3), 257–284 (1999)
Comfort, W.W., Ross, K.A.: Topologies induced by groups of characters. Fund. Math. 55, 283–291 (1964)
Engelking, R.: General Topology. Sigma Series in Pure Mathematics. Heldermann Verlag, Berlin (1989)
Gabriyelyan, S.: A locally quasi-convex abelian group without Mackey topology. In: Proceedings of the American Mathematical Society. https://doi.org/10.1090/proc/14020
Glicksberg, I.: Uniform boundedness for groups. Can. J. Math. 14, 269–276 (1962)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis I. Die Grüundlehren der Mathematischen Wissenschaften. Springer, Berlin (1963)
Margalef, J., Outerelo, E., Pinilla, J.L.: Topología. Alhambra, Granada (1980)
Martín-Peinador, E.: Normality on Topological Groups. Mathematical Contributions in honor of Juan Tarrés (Spanish), pp. 287–293. Univ. Complut. Madrid, Fac. Mat., Madrid (2012)
Martín-Peinador, E., Tarieladze, V.: Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks. RACSAM 110(2), 667–679 (2016)
Oxtoby, J.C.: Cartesian products of Baire spaces. Fund. Math. 49, 157–166 (1960/1961)
Remus, D., Trigos-Arrieta, F.J.: Abelian groups which satisfy Pontryagin duality need not respect compactness. Proc. Am. Math. Soc. 117(4), 1195–1200 (1993)
Stone, A.H.: Paracompact and product spaces. Bull. Am. Math. Soc. 54, 977–982 (1948)
Trigos-Arrieta, F.J.: Every uncountable Abelian group admits a non-normal group topology. Proc. Am. Math. Soc. 122(3), 907–909 (1994)
Willard, S.: General Topology. Addison-Wesley, Reading (1970)
Acknowledgements
The authors are deeply grateful to the referees who have read carefully the first draft of the manuscript, and have done very valuable comments. In particular, the last section in this version was originated by direct suggestions of one of the referees.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor María Teresa Lozano Imízcoz on the occasion of her 70th birthday.
E. Martín-Peinador acknowledges the financial support of the Spanish AEI and FEDER UE funds (Grant MTM2016-79422-P).
Rights and permissions
About this article
Cite this article
Martín-Peinador, E., Pérez Valdés, V. A class of topological groups which do not admit normal compatible locally quasi-convex topologies. RACSAM 112, 867–876 (2018). https://doi.org/10.1007/s13398-018-0507-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-018-0507-y