Pseudocompact refinements of compact group topologies

Non-Abelian Pseudocompact Groups

“…The result is well known (see for example [40] (6.1)) when G, hence also H, is compact. Denoting as in 1.3(b) by G the Weil completion of (an arbitrary) totally bounded group G, we have in the present case G/H = G/H (see also in this connection [93] …”

“…Comfort and Remus [40] (5.5) responded positively for many (non-metrizable) compact abelian groups K, including for example those which are connected, or torsion, or which satisfy cf(w(K)) > ω. Later Comfort and Galindo [46] 2)), in the compact abelian case K with w(K) = α > ω, there are 2 2 2 α -many pseudocompact group refinements of weight 2 α .…”

“…It is known that every non-metrizable compact group K which is either abelian or connected is product-like (see for example [40] (5.4) or [74] (1.11) for the abelian case, [40] (proof of 6.5) or [73] for the connected case). This allowed the authors of [70,[73][74][75] to conclude that for every such group K the set Ω(K) of dense ω-bounded subgroups of K satisfies |Ω(K)| ≥ |K|.…”

“…In the proof for connected groups ( [75] (5)) the authors used [4] ((4.2), (4.3)). The paper [40] was cited in [74] (2.5), but not in [75]. The authors of [70,74,75] inadvertently failed to note that already in [40] (proof of (6.5)) the following more detailed result had been obtained in a more direct way (see also [40] (5.4) for a relevant result in the abelian context):…”

Pseudocompact topological group refinements of maximal weight

“…This question is posed explicitly by the authors of [5], who obtain a positive answer in many special cases. Our goal in this paper is to answer this question affirmatively in full generality (Theorem 5.1), and to also show that every torsion-free pseudocompact Abelian group G with ω < w(G) ≤ |G| = |G| ω admits a pseudocompact group refinement of maximal weight (Corollary 5.3).…”