Connected, not separably connected complete metric spaces
Тарас Онуфриевич Банах, Myroslava I. Vovk, Wójcik, M. R. · arXiv (Cornell University) · 2009
In a separably connected space any two points are contained in a separable connected subset. We show a mechanism that takes a connected bounded metric space and produces a complete connected metric space whose separablewise components form a quotient space isometric to the original space. We repeatedly apply this mechanism to construct, as an inverse limit, a complete connected metric space whose each separable subset is zero-dimensional.