Dense embeddings in pathwise connected spaces

Abstract

This paper is devoted to the problem of finding those T 1-spaces (Hausdorff spaces) which are densely embeddable in a pathwise connected T 1-space (Hausdorff space). In particular, we prove that a countable first countable Hausdorff space (with more than one point) is pathwise connectifiable (i.e., it can be densely embedded in a pathwise connected Hausdorff space) if and only if it has no isolated points. Moreover some examples are given to show that a Hausdorff space which can be densely embedded in a connected Hausdorff space may fail to be pathwise connectifiable.