Transversality on locally pseudocompact groups

Fucai LIN; Zhongbao TANG

doi:10.1007/s11464-021-0940-7

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 LIN ¹^,²^,^†
, Zhongbao TANG ¹

Author information +

History +

PDF (269KB)

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

Cite this article

Download citation ▾

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

登录浏览全文

4963

注册一个新账户忘记密码

References

Publishing order | Descend order by publishing year | Descend order by cited within

[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

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

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

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

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

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

[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

[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

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

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

[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

[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

[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