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Comfort, W.W. and J. van Mill, Some topological groups with, and some without, proper dense subgroups, Topology and its Applications 41 (1991) 3-15. Continuing earlier investigations into the question cf the existence of a proper dense subgroup of a given topological group, the authors obtain some positive and some negative results, as follows: (a) Every non-degenerate connected Abelian group contains a proper dense subgroup. (b) Every infinite pseudocompact group contains a proper dense subgroup. (c) If G is a totally bounded Abelian group with wG = ICI= a! > w, and if in addition G is torsion-free (or more generally, if IG/tor Gl = r), then G has a proper dense subgroup. (d) For every strong limit cardinal a such that cf(a) = o there is a totally bounded Abelian torsion group G such that wG = IGi = a and G has no proper dense subgroup. Among the questions left unsettled is this: Can the conditions in (d) on a be relaxed or even omitted? .!Y.I;::Yz~ Totally bounded group, ploprr dense subgroup, torsion-free group, pseudocompact group, connected group. AMS(MOS) Subj. Class.: 54AO5,201(45,22CO5. Notation and cowentions Each ordinal number (in particular, each cardinal number) is i entified with its set of ordinal predecessors. That is, for each ordinal number 5 we write t=b?: rl<o=bl: rl~5~* The smallest infinite cardinal is denoted w; the symbols a, /3, y, K denote in cardinals. 0166-8641/91/$03.50 @ 1991-Elsevier Science Publishers R.\i. (North-Hoiiand! 4 W. W. Comfort, J. van Mill The cardinality of a set X is written 1x1. The weight and density of a space X are written WX and dX, respectively. The symbols Z, R and U denote the integers, the reals and the circle, respectively; in each case the usual algebraic operations and the usual topology are understood. In dealing with T we use multiplicative notation and we denote the identity by 1; for every other Abelian group we use additive notation and we denote the identity by 0. The torsion subgroup of an Abelian group G is written tG, and 4 : G + G/ tG is the canonical homomorphism. The topological groups we consider are all assumed to satisfy the Hausdorff separation axiom. As is well known (see for example [ 11,8.4], this condition guarantees that our groups are completely regular spaces, i.e., Tychonoff spaces.