Countable networks on Malykhin’s maximal topological group. (English) Zbl 07779893

Márquez, Edgar

Appl. Gen. Topol. 24, No. 2, 239-246 (2023).

The author presents a solution to the following problem. Does every countable and non-discrete topological (abelian) group have a countable network with infinite elements? In fact, the author shows that no maximal topological space allows for a countable network with infinite elements. As a result, he answers the question in the negative. The article also focuses on Malykhin’s maximal topological group constructed in 1975 and establishes some unusual properties of countable networks on this special group \(G\).

Reviewer: Zhangyong Cai (Nanning)

MSC:

22A05	Structure of general topological groups
54H11	Topological groups (topological aspects)

Keywords:

countable network; resolvable; linear; \(P\)-point; \(P\)-space

References:

[1]	A. V. Arhangel’skii and M. G. Tkachenko, Topological Groups and Related Structures, Atlantis Studies in Mathematics, Vol. I, Atlantis Press and World Scientific, Paris-Amsterdam, 2008. https://doi.org/10.2991/978-94-91216-35-0 · Zbl 1323.22001 · doi:10.2991/978-94-91216-35-0
[2]	E. K. van Douwen, The Integers and Topology, in: Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, Eds.), Elsevier Science Publ. B. V. (1984), 111-167. https://doi.org/10.1016/B978-0-444-86580-9.50006-9 · Zbl 0546.00022 · doi:10.1016/B978-0-444-86580-9.50006-9
[3]	D. H. Fremlin, Consequences of Martin’s Axiom, Cambridge University Press, Cambridge, 1984. https://doi.org/10.1017/CBO9780511896972 · Zbl 0551.03033 · doi:10.1017/CBO9780511896972
[4]	E. Márquez and M. Tkachenko, D-independent topological groups, Topology Appl. 300 (2021), 107761. https://doi.org/10.1016/j.topol.2021.107761 · Zbl 1497.22001 · doi:10.1016/j.topol.2021.107761