Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups

Abstract

2.2. COROLLARY. Let K be a compact topological group with w(K) = a > ω. Then (a) K contains a dense, countably compact subgroup G such that \G = (log(α)); and (b)m(lθ ω. From the Hewitt-Marczewski-Pondiczery theorem (cf. [18] (2.3.15)) there is a dense subset D of {0, l} such that |Z>| | = 2. For every space X in which each point is a G8 and 2 < \X < 2 ω we have \X = 2 and P(X) » A so that P(JT) * P ( ( * T ) P ( ( P ( J T ) Π « P{D). A topological space ^ is said to be a Baire space if the countable intersection of dense, open subsets of X is dense in X. 2.4. LEMMA, (a) If X is a compact (Hausdorff) space, then PX is a Baire space. (b) A G8-dense subspace of a Baire space is a Baire space. Proof, (a) The set 3S of compact Gδ-sets of X is a base for the topology of PX. Let B e SS and let {Un: n < ω} be a sequence of dense, open subsets of PX with Un 3 Un+V We show (Γ Un) n B Φ 0. Choose x 0 e ί Π ί / 0 and then Bo e ^ such that x0 e 5 0 c 5 Π ί/0 and recursively, if xk and 5 Λ have been defined, choose xk+ι^ BkΠ Uk+ι and then Bk+1 e ^ so that xA:+1 e ^A:+i c ^ n ^4+i' s i n c ^ each Bk is X-compact we have 0 * Πrt 5M c (ΠM C/J Π 5, as required. (b) Let Y be Gδ-dense in the Baire space X, let U be a non-empty open subset of F and let Un be dense and open in Y. There are open sets U9 Un in X such that i7 = U Π y and Un = ί/Λ Π y. Since X is Baire and Un is dense in X we have (Γ)nUn) n U Φ 0, and since Yis Gδ-dense in X we then have