How many ω-bounded subgroups?

A topological space is said to be ω-bounded if each of its countable subsets has compact closure. It has been shown recently by Itzkowitz and Shakhmatov that for every compact Abelian group G of uncountable weight, and for every compact connected group of G of uncountable weight, the set Ω(G) of dense ω-bounded subgroups of G satisfies Ω(G) ⩾ G. These authors asked whether 0977 their estimate Ω(G) ⩾ G may be improved to Ω(G) = 2G for some or all such G. In the 0977 0605 V 3 present paper we answer this question affirmatively for all compact groups G which are either Abelian or connected and which satisfy in addition the condition w(G) = (w(G))ω. We show also that every compact group G with ω(G) ⩾ log((2′)+) satisfies Ω(G) > 2′.