Protasov, I. V. 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. Reviewer: M.I.Ursul (Kishinev) Cited in 2 ReviewsCited in 11 Documents MSC: 22A05 Structure of general topological groups 54H11 Topological groups (topological aspects) Keywords:topological group; infinite totally bounded subset; nonclosed discrete subset Citations:Zbl 0696.22003; Zbl 0727.22002 PDFBibTeX XMLCite \textit{I. V. Protasov}, Math. Notes 55, No. 1, 1 (1994; Zbl 0836.22003); translation from Mat. Zametki 55, No. 1, 150--151 (1994) Full Text: DOI References: [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). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.