Local properties of topological spaces and remainders in compactifications (Q2322494)

For topological properties \(\mathcal{P}\) and \(\mathcal{Q}\), a space \(X\) is \((\mathcal{P,Q})\)-structured if \(X\) has a subspace \(W\in\mathcal{P}\) such that for each open neighbourhood \(U\) of \(W\) in \(X\) we have \(X\setminus U\in\mathcal{Q}\). Some general results such as the following are presented: if \(\mathcal{P}\) and \(\mathcal{Q}\) are closed-hereditary classes of spaces such that \(Y\) is a remainder of \(Z\in\mathcal{P}\ \Rightarrow\ Y\in\mathcal{Q}\) and a finite union of members of \(\mathcal{P}\) is again in \(\mathcal{P}\) provided each is closed in the union then every remainder of a space that is locally in \(\mathcal{P}\) is \((\mathcal{K},\mathcal{Q})\)-structured, where \(\mathcal{K}\) is the class of compacta. These results are then applied to specific instances of \((\mathcal{P},\mathcal{Q})\), for example (Lindelöf \(p\), Lindelöf \(p\)) and (metrisable, charming). Similar generalities lead to: if \(X\) is locally metrisable (and locally separable) with a homogeneous remainder in some compactification then \(Y\) is charming (Lindelöf \(p\); and if, further, \(X\) is also nowhere locally compact then \(X\) is separable and metrisable), and if \(X\) is locally metrisable and \(Y\) a remainder of \(X\) such that each point of \(Y\) is \(G_\delta\) in \(Y\) then \(\vert Y\vert \le 2^\omega\).