Filters and pathwise connectifications (Q2764720)

A Hausdorff space \(X\) is pathwise connectifiable if there exists a Hausdorff space \(Y\) in which \(X\) can be embedded densely; \(Y\) is then said to be a pathwise connectification of \(X\). The authors study the existence of pathwise connectifications of \(X\cup\{p\}\) where \(p\) is a free open filter on \(X\) and a local base at \(p\) is of the form \(\{U\cup\{p\}:U\in p\}\). The main results involve conditions on the open filter \(p\) in the cases in which \(X\) is either the space of rationals or the real line.