Vol. 49, No. 1, 1973

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Countably compact groups and finest totally bounded topologies

W. Wistar (William) Comfort and Victor Harold Saks

Vol. 49 (1973), No. 1, 33–44

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 0, then G is dense in a countably compact group H such that |H|n0. 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|n0.

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| < 0.

Mathematical Subject Classification 2000
Primary: 22A05
Received: 7 July 1972
Published: 1 November 1973
W. Wistar (William) Comfort
Victor Harold Saks