Refining connected topological group topologies on Abelian torsion groups

Neither first countable nor Cech-complete spaces are maximal Tychonoff connected

“…The following technical lemma, which is fundamental to the main theorems, is a modification of Proposition 4.2 of [16]. …”

“…Even so, the folklore suspicion is that such spaces do not exist. The first steps of an attempted proof were taken in [10], where it was shown that there exists a connected group topology on the reals R which is stronger than the usual one (see also Example 2.10 of [1]); in [16] stronger connected group topologies were constructed for certain Abelian topological groups.…”

Continuity properties of discontinuous homomorphisms and refinements of group topologies

“…The general problem of finding a finer topological group topology on a topological abelian group G preserving some of the properties of G was considered by several authors in [1,[3][4][5][6][7]11] and [14,15,17,18]. This study has been mainly focused on the preservation of pseudocompactness or connectedness.…”