Proper Pseudocompact Extensions of Compact Abelian Group Topologies

Non-Abelian Pseudocompact Groups

“…(a) Since a pseudocompact normal space is countably compact ([1] (3.10.21), [6] (3.D.2)) and a countably compact metric space is compact [1] (4.1.17), we have, as noted frequently in the literature ([36] (4.5(a)), [37] (3.1), [34] (2.4, 3.6)): * every pseudocompact group G with w(G) ≤ ω is both r-and s-extremal. This explains the occurrence of the hypothesis "w(G) > ω" (equivalently: "G is non-metrizable") in many of the theorems cited below.…”

“…Comfort and Robertson [37] (3.2(b)) showed that each non-metrizable compact abelian group K maps by a continuous epimorphism onto a group of the form M (ω + ) with M a compact subgroup of T. From this they determined [37] (3.4) that such K is not r-extremal.…”

Pseudocompact topological group refinements of maximal weight

“…The technique we use to treat the case r 0 (G) = |G| > c of Theorem 5.1 is new. This paper is self-contained but where possible we invoke and adapt the technique of [6] as developed in [5] to handle the cases r 0 (G) < |G| or |G| = c.…”

Extremal pseudocompact Abelian groups are compact metrizable

“…G has no proper dense pseudocompact subgroup]. Early formulations of these notions appeared in [6], [7]. From the fact that a pseudocompact space of countable weight is compact and metrizable it follows readily (as in [7, 2.3]) that every pseudocompact group of countable weight is both r-extremal and s-extremal.…”