IMPOSING PSEUDOCOMPACT GROUP TOPOLOGIES ON ABELIAN-GROUPS

The least cardinal lambda such that some (equivalently: every) compact group with weight alpha admits a dense, pseudocompact subgroup of cardinality lambda is denoted by m(alpha). Clearly, m(alpha) less-than-or-equal-to 2alpha. We show: THEOREM 3.3. Among groups of cardinality gamma, the group +(gamma)Q serves as a ''test space'' for the availability of a pseudocompact group topology in this sense: If m(alpha) less-than-or-equal-to gamma less-than-or-equal-to 2alpha then +(gamma)Q admits a (necessarily connected) pseudocompact group topology of weight alpha greater-than-or-equal-to omega (and also a pseudocompact group topology of weight log gamma). THEOREM 4.12. Let G be Abelian with Absolute value of G = gamma. If either m(alpha) less-than-or-equal-to alpha and m(alpha) less-than-or-equal-to r0(G) less-than-or-equal-to gamma less-than-or-equal-to 2alpha, or alpha < omega and alpha(omega) less-than-or-equal-to r0(G) less-than-or-equal-to 2alpha, then G admits a pseudocompact group topology of weight alpha. THEOREM 4.15. Every connected, pseudocompact Abelian group G with wG = alpha greater-than-or-equal-to omega satisfies r0(G) greater-than-or-equal-to m(alpha). THEOREM 5.2(b). If G is divisible Abelian with 2r0(G) less-than-or-equal-to gamma, then G admits at most 2gamma-many pseudocompact group topologies. THEOREM 6.2. Let beta = alpha(omega) or beta = 2alpha with beta greater-than-or-equal-to alpha, and let beta less-than-or-equal-to gamma < kappa less-than-or-equal-to 2beta. Then both +(gamma)Q and the free Abelian group on gamma-many generators admit exactly 2kappa-many pseudocompact group topologies of weight kappa. Of these, some kappa+-many form a chain and some 2kappa-many form an anti-chain.