Two remarks on weaker connected topologies (Q2761031)

A topological space \((X,\mathcal T_0)\) condenses onto a space \((X,\mathcal T_1)\) if there is a topology \(\mathcal T_2\subset \mathcal T_0\) on \(X\) such that \((X,\mathcal T_2)\) and \((Y,\mathcal T_1)\) are homeomorphic. Under the assumption that the real line cannot be covered by \(\aleph_1\) nowhere dense sets, the authors prove that no discrete sum of \(\aleph_1\) zero-dimensional compact Hausdorff spaces of which at least two are nonempty condenses onto a connected Hausdorff space. This solves, in the negative, two problems formulated by \textit{M. G. Tkachenko, V. V. Tkachuk, V. V. Uspenskij} and \textit{R. G. Wilson} [Commentat. Math. Univ. Carol. 37, No. 4, 825-841 (1996; Zbl 0886.54034)].