|
The first main result formalizes
the general principle that each totally bounded group G is dense in some group
H, not much larger than G, in which every subset of small cardinality has
a complete accumulation point. For example: If G is totally bounded and
|G| = n ≧ℵ0, then G is dense in a countably compact group H such that
|H|≦ nℵ0. A corollary: If K is an infinite compact group with weight not
exceeding 2n, then K contains a dense, countably compact subgroup H with
|H|≦ nℵ0.
The following results are given in §2: If t is the finest totally bounded topological
group topology on an infinite Abelian group G, then every subgroup of G is t-closed
and (G,t) is not pseudocompact (both conclusions can fail for G non-Abelian); a
closed subgroup of a pseudocompact group need not be pseudocompact;
if {(Gi,ti} : i ∈ I} are nontrivial Abelian groups with their finest totally
bounded topologies and (G,∼ Z) is their product, then 𝒯 = t if and only if
|I| < ℵ0.
|
