Which spaces have a coarser connected Hausdorff topology? (Q1862085)

Some answers to the question in the title are considered; that is to say, the authors seek necessary and sufficient conditions in order that there exists a continuous bijection (or condensation) from a space \((X,\tau)\) to a connected Hausdorff space. Section 2 of the paper is dedicated to obtaining some necessary conditions for the existence of a weaker connected topology while in Sections 4 and 5 some sufficient conditions are given using the somewhat technical concepts of \(\kappa\)-epoxicity and \(\kappa\)-super-epoxicity. As corollaries to the technical results, it is shown that every cardinal has a weaker connected Hausdorff topology (for the cardinal \(\omega_1\) this was first shown by \textit{S. Christodoulou} [Quest. Answers Gen. Topology 10, No. 2, 143-148 (1992; Zbl 0782.54002)]) and that \((2^{\omega})^{+}+\omega\) is the smallest non-compact ordinal with no such weaker topology. Section 6 considers extensions of the discrete space \(\omega\) and the main theorem of this section states that a locally countable, locally compact extension of \(\omega\) of cardinality \(\omega_1\) has a weaker connected Hausdorff topology if \(\omega_1< {\mathfrak p}\). The final section considers spaces between \(\omega\) and \(\beta\omega\) and includes an interesting example of a Hausdorff space which can be condensed onto a connected Hausdorff space but whose semiregularization cannot be so condensed.