Partially ordered set of zero-dimensional one-point extensions of a topological space (Q6150790)

For a zero-dimensional topological space \(X\), let \(\mathcal{E}(X)\) be the poset of one-point extensions of \(X\) and \(\mathcal{E}_0(X)\) the subposet of zero-dimensional one-point extensions of \(X\). The Banaschewski compactification of \(X\) is the largest zero-dimensional compactification of \(X\) and is denoted \(\zeta X\). Among many interesting results on \(\mathcal{E}_0(X)\), the author shows if \(Y \leq Z \leq W\) in \(\mathcal{E}(X)\) and \(Z \in \mathcal{E}_0(X)\), neither \(Y\) nor \(W\) are necessarily in \(\mathcal{E}_0(X)\). Each \(Y \in \mathcal{E}_0(X)\) can be associated with a clopen bornology on \(X\). \(Y \in \mathcal{E}_0(X)\) is the supremum of the elements in \(\mathcal{E}_0(X)\) which are covered by \(Y\). \(\mathcal{E}_0(X)\) has a minimal element if and only if \(X\) is locally compact. \(\mathcal{E}_0(X)\) is a lower semilattice and for any \(Y \in \mathcal{E}_0(X)\), \(\{Z \in \mathcal{E}_0(X) : Y \leq Z\}\) is a complete lower semilattice. For \(Y \in \mathcal{E}_0(X)\), the Banaschewski compactification \(\zeta Y\) is described as a quotient of \(\zeta X\). Connections to locally \(\mathbb{N}\)-compact and locally Lindelöf spaces are also considered.