Coarser connected metrizable topologies (Q989100)

The main result of this paper is that whenever \(\tau\) is a non-compact metrizable topology on a set \(X\) of weight at least \(\mathfrak c\), then there is a connected metrizable topology \(\mu\subseteq\tau\) on \(X\). It is known that if \(w(X)<\mathfrak c\), then no such connected regular topology \(\mu\) need exist. The principal theorem generalizes results of a) \textit{G. Gruenhage, V. V. Tkachuk} and the reviewer [Topology Appl. 120, No.~3, 365--384 (2002; Zbl 1007.54029)] who showed the existence of a connected Hausdorff topology \(\mu\subseteq\tau\); b) \textit{W. G. Fleissner, J. Porter} and \textit{J. Roitman} [Topology Appl. 142, No.~1--3, 131--157 (2004; Zbl 1068.54002)] who showed that such a metrizable topology \(\mu\) exists whenever \(\tau\) is zero-dimensional; and c) \textit{I. Druzhinina} [Houston J. Math. 30, No.~3, 751--766 (2004; Zbl 1059.54022)] who showed that such a metrizable topology \(\mu\) exists if the extent of \((X,\tau)\) is attained.