Universal spaces for some families of rim-scattered spaces (Q1202795)

The families of rim-scattered spaces alluded to in the title are denoted \(R^ k_{text{c}}(\alpha)\) and \(R^ k_{\text{lc}}(\alpha)\) respectively. These spaces are to have a base \(\mathcal B\) of open sets such that the boundary of every element of \(\mathcal B\) has scattered height at most \(\alpha\) and a compactification (in the case of \(R^ k_{\text{lc}}(\alpha)\) a locally compact extension) of scattered height at most \(\alpha+k+1\) (this is deduced from the context; in the paper the parameter \(k\) does not appear in the definitions of \(R^ k_{\text{c}}(\alpha)\) and \(R^ k_{\text{lc}}(\alpha)\)). In this rather technical paper the author constructs universal spaces for the classes \(R^ k_{\text{lc}}(\alpha)\); in fact one can, given a subfamily of \(R^ k_{\text{lc}}(\alpha)\) of size at most \(\mathfrak c\), find a space that contains an almost disjoint (in some sense) family of copies of the elements of the family.