Connectifying some spaces

A Hausdorff space X is called (countably) connectifiable if there exists a connected Hausdorff space Y (with |Y⊮X| ⩽ ω; respectively) such that X embeds densely into Y. We prove that it is consistent with ZFC that there exists a regular dense in itself countable space which is not countably connectifiable giving thus a partial answer to Problem 3.9 of Watson and Wilson (1993). On the other hand we show that Martinʹs axiom implies that every countable dense in itself space X with πω(X) < 2ω is countably connectifiable. We also establish that a separable metrizable space without open compact subsets can be densely embedded in a metric continuum.