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Discrete subsets of topological groups. (English. Russian original) Zbl 0836.22003

Math. Notes 55, No. 1, 101-102 (1994); translation from Mat. Zametki 55, No. 1, 150-151 (1994).
It is proved that if a topological group contains an infinite totally bounded subset, then it contains a nonclosed discrete subset. This result gives, in particular, a positive answer to a question of E. K. van Douwen [Topology Appl. 34, 69-91 (1990; Zbl 0696.22003)]. Another proof was given by the reviewer [Bul. Acad. Ştiinţe Repub. Mold. Mat. 1991, No. 3, 67-69 (1991)]; see also K. P. Hart and J. van Mill [J. Pure Appl. Algebra 70, 73-80 (1991; Zbl 0727.22002)].
Problem: Construct in the ZFC a countable nondiscrete group, all of whose discrete subsets are closed.

MSC:

22A05 Structure of general topological groups
54H11 Topological groups (topological aspects)
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[1] E. K. van Douwen, ”The maximal totally bounded group topology on G and the biggest minimal G-space for Abelian groups G,” Topology Appl.,34, No. 1, 69–91 (1990). · Zbl 0696.22003 · doi:10.1016/0166-8641(90)90090-O
[2] M. I. Ursul, ”On a problem of van Douwen,” Izv. Akad. Nauk Respub. Moldova Mat., No. 3, 67–69 (1991).
[3] V. I. Malykhin, ”Extremally disconnected topological groups,” Usp. Mat. Nauk,34, No. 6, 59–66 (1979). · Zbl 0426.22002
[4] V. I. Malykhin, ”Extremally disconnected and nearly extremally disconnected groups,” Dokl. Akad. Nauk SSSR,220, No. 1, 27–30 (1975).
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