Transversality on locally pseudocompact groups

Fucai LIN; Zhongbao TANG

doi:10.1007/s11464-021-0940-7

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Front. Math. China ›› 2021, Vol. 16 ›› Issue (3) : 771-782. DOI: 10.1007/s11464-021-0940-7

RESEARCH ARTICLE

Transversality on locally pseudocompact groups

Fucai LIN¹^,² ,
Zhongbao TANG¹

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Abstract

Two non-discrete Hausdorff group topologies $τ$ and $δ$ on a group G are called transversal if the least upper bound $τ ∨ δ$ of $τ$ and $δ$ is the discrete topology. In this paper, we discuss the existence of transversal group topologies on locally pseudocompact, locally precompact, or locally compact groups. We prove that each locally pseudocompact, connected topological group satisfies central subgroup paradigm, which gives an affrmative answer to a problem posed by Dikranjan, Tkachenko, and Yaschenko [Topology Appl., 2006, 153:3338-3354]. For a compact normal subgroup K of a locally compact totally disconnected group G, if G admits a transversal group topology, then G/K admits a transversal group topology, which gives a partial answer again to a problem posed by Dikranjan, Tkachenko, and Yaschenko in 2006. Moreover, we characterize some classes of locally compact groups that admit transversal group topologies.

Keywords

Transversal group topology / locally pseudocompact group / locally compact group / locally precompact / connected space / central topological group

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Fucai LIN, Zhongbao TANG. Transversality on locally pseudocompact groups. Front. Math. China, 2021, 16(3): 771‒782 https://doi.org/10.1007/s11464-021-0940-7

References

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[1]	Agrawal M R, Kanpur U B. On existence of finite universal Korovkin sets in the centre of group Algebra. Monatsh Math, 1997, 123: 1–20 CrossRef Google scholar

[2]	Arhangel'skii A, Tkachenko M. Topological Groups and Related Structures. Atlantis Stud Math, Vol 1. Paris/Hackensack: Atlantis Press/World Scientific, NJ, 2008 CrossRef Google scholar

[3]	Birkhoff G. On the combination of topologies. Fund Math, 1936, 26: 156–166 CrossRef Google scholar

[4]	Dikranjan D. Recent advances in minimal topological groups. Topology Appl, 1998, 126: 149–168 CrossRef Google scholar

[5]	Dikranjan D, Tkachenko M, Yaschenko I. On transversal group topologies. Topology Appl, 2005, 153: 786–817 CrossRef Google scholar

[6]	Dikranjan D, Tkachenko M, Yaschenko I. Transversal group topologies on non-abelian group. Topology Appl, 2006, 153: 3338–3354 CrossRef Google scholar

[7]	van Douwen E K. The weight of a pseudocompact (homogeneous) space whose cardinality has countable cofinality. Proc Amer Math Soc, 1980, 80(4): 678–682 CrossRef Google scholar

[8]	Engelking R. General Topology (revised and completed ed). Berlin: Heldermann Verlag, 1989

[9]	Grosser S, Moskowitz M. On central topological groups. Trans Amer Math Soc, 1967, 127: 317–340 CrossRef Google scholar

[10]	Hofmann K H, Morris S. The Structure of Compact Groups. Berlin: de Gruyter, 2013 CrossRef Google scholar

[11]	Peyrovian M R. Maximal compact normal subgroups. Proc Amer Math Soc, 1987, 99(2): 389–394 CrossRef Google scholar

[12]	Prodanov I R, Stoyanov L N. Every minimal abelian group is precompact. Dokl Bulg Acad Sci, 1984, 37: 23–26

[13]	Stephenson R M. Minimal topological groups. Math Ann, 1971, 192: 193–195 CrossRef Google scholar

[14]	Tkacenko M G, Tkachuk V V, Wilson R G, Yaschenko I. No submaximal topology on a countable set is T1-complementary. Proc Amer Math Soc, 2000, 128(1): 287–297 CrossRef Google scholar

[15]	Weil A. Sur les Espaces à Structure Uniforme et sur la Topologie Génénrale. Publ Math de l'Université Strasbourg. Paris: Hermann $ Cie, 1937

[16]	Zelenyuk E, Protasov I. Complemented topologies on abelian groups. Sibirsk Mat Zh, 2001, 42(3): 550-560 (in Russian); Sib Math J, 2001, 42(3): 465–472 CrossRef Google scholar