Dense embeddings in pathwise connected spaces
A (pathwise) connectification of a space \(X\) is a (pathwise) connected topological space \(Y\) containing a dense (homeomorphic) copy of \(X\). \textit{S. Watson} and the reviewer [Houston J. Math. 19, No.~3, 469-481 (1993; Zbl 0837.54012))] have shown that the non-existence of isolated points is a necessary and sufficient condition for the existence of a \(T_1\) connectification of a \(T_1\)-space and a \(T_2\) connectification of a countable Hausdorff space. This paper studies the existence of pathwise connectifications and it is shown that the same condition (of being dense-in-itself) characterizes the existence of \(T_1\) pathwise connectifications of \(T_1\)-spaces and \(T_2\) pathwise connectifications of countable, first countable Hausdorff spaces. Examples are given of a countable Hausdorff space and a subspace of the real line having \(T_2\) connectifications, but which have no \(T_2\) pathwise connectification. A number of open problems are posed.
scientific article; zbMATH DE number 1340406