\(\omega\)-connectifications and product spaces (Q2703103)

A space \(X\) is said to be \(\omega\)-connectifiable (respectively, 1-point connectifiable) if it can be densely embedded in a connected Hausdorff space \(Y\) in such a way that \(Y\smallsetminus X\) has cardinality at most \(\aleph_0\) (respectively, cardinality at most 1). The aim of this paper is to prove the following theorem: Let \(\{X_\alpha:\alpha\in A\}\) be a family of Hausdorff spaces whose components are open; then, (1) \(X\) is \(\omega\)-connectifiable if and only if it has no proper open subsets which are \(H\)-closed, and (2) \(X\) is 1-point connectifiable if and only if either \(X\) is connected or there is some \(X_\alpha\) which has no \(H\)-closed components.