On the connectification of a space by a countable point set

In this note we define a rather pathological connectedness property of Hausdorff spaces which is stronger than ordinary connectedness. We obtain a few basic properties of such spaces and derive a method for constructing them. It turns out that countable Hausdorff spaces having the connectedness property are as easily constructed as uncountable ones. Hence we have still another method for constructing countable connected Hausdorff spaces.