Every locally connected functionally Hausdorff space is \(c\)-resolvable (Q2253722)

All spaces are assumed \(T_1\). For a cardinal \(\kappa\), a space \(X\) is \(\kappa\)-resolvable if it contains \(\kappa\) mutually disjoint dense subsets and \(\kappa\)-retodic if it may be partitioned by a family \(\{D_\xi\}_{\xi\in\kappa}\) of dense subsets such that \(X\setminus D_\xi\) is totally disconnected for each \(\xi\). Suppose that \(X\) is locally connected and for each connected open set \(U\subset X\) there is a \(\kappa\)-retodic space \(Y_U\) and a non-constant map \(f:U\to Y_U\): then \(X\) is \(\kappa\)-resolvable. In particular functionally Hausdorff, locally connected spaces are \(\mathfrak c\)-resolvable.