A note on everywhere dense subgroups

Abstract

In the first issue of these Proceedings [1]2 Hsien-Chung Wang has shown that in any separable nondiscrete compact and group there is always an uncountable everywhere dense proper subgroup. Since any countable subset of a nondiscrete T1space is first category, this result is implied by each of the following conjectures, in which X is an arbitrary topological group and G any proper subgroup of X: (C1) if X is uncountable, G lies in some uncountable proper subgroup; (C2) if X is category, G lies in some category proper subgroup. Using Wang's methods we here verify these conjectures for certain special cases and also establish his theorem with locally compact and metric replaced by second category and Hausdorff. Let X be any group with e the identity element and let 4 be a family of functions on X to X including the identity function i(x) =x. Let (M denote the class of all non-null proper 4-subgroups G in X, '?-subgroup meaning a subgroup G such that ?>(G) CG for every 4Cb. For each GEC take Y1(G) to consist of all pairs (1; g) with gEG, and for n >2 let Y.(G) be composed of all 2n-tuples y of the form y = (n; gl, * * * , gn; cf1, * * * , On-1) with giGG and CiGb. Set YG = U1 Yn(G) and let J'G on XX YG to X be defined by JVG(X, y) =g when y = (1; g) and VPG(X, y) =g1c1(x)g2q2(x) ... gn_10n_l(X)gn when yE Yn(G) for n > 2. For fixed G in (M and x in X we write G(x) for the set 4/G(X; YG) and note that G(x) is always a semigroup in X and that G(x)3x since YG3 (2; e, e; i). Moreover, G(x) DIPIG(X, Y1(G)) =G; hence G(g) =G when gEG, since /i(g)EG for gEG and ciCb. For each qEX set SG(q)= [x G(x)3q] and define SG(Q) for any set Q in X to be U [SG(q) | qEQ]. Finally, for any fixed element Go of 5 and any non-null subset P of the complement X-Go of Go, let (5(Go, P) be all G in (M which contain Go and are disjoint to P. We note that by an obvious application of the Hausdorff Maximality Principle (5(Go, P) always contains at least one maximal element.