In what follows, a \(P\)-space is a topological space on which all \(G_\delta\) subsets are open. A topological space \(X\) is said to be crowded if it has no isolated points, and it is scattered (or dispersed) if every non-empty subspace \(Y\) of \(X\) contains a point which is isolated in \(Y\). We assume the reader is familiar with the derived set, Cantor-Bendixson derivative and the Cantor-Bendixson rank (denoted by \(N(A)\)) of a subset \(A\) of a topological space \(X\). In particular, the reader may easily recall that if \(X\) is a scattered space then \(N(X)\) is the least ordinal \(\alpha\) such that the \(\alpha\)-th Cantor-Bendixson derivative \(X^{(\alpha)}\) is empty. It follows that \(X^{(\beta)} \neq \emptyset\) whenever \(\beta < N(X)\) (for dispersed \(X\)) and, if \(X\) is a scattered space of size \(\aleph_1\), then \(N(X) < \omega_2\).
The weight of a topological space \(X\) is the minimal cardinality for a base of \(X\).
The paper under review is a continuation of W. Bielas et al. [Topology Appl. 312, Article ID 108064, 18 p. (2022; Zbl 1507.54016)] – on which the authors have studied crowded \(P\)-spaces of size and weight \(\aleph_1\) – now focusing on scattered \(P\)-spaces of weight \(\aleph_1\). A general idea that some proofs on scattered \(P\)-spaces of weight \(\aleph_1\) and countable Cantor-Bendixson derivative are similar to proofs concerning scattered separable metric spaces is explored and permeates the whole paper.
The paper begins with the easy, instructive observation that any scattered space of weight at most \(\aleph_1\) has size not larger than \(\aleph_1\). A large number of results are established. For instance, it is shown that any regular scattered \(P\)-space of weight \(\aleph_1\) can be embedded into \(\omega_2\) (with the order topology). As a corollary of the previous result, a regular scattered \(P\)-space of weight \(\aleph_1\) has a scattered compactification of cardinality \(\aleph_1\).
The paper finishes with some remarks concerning \(P\)-spaces with uncountable Cantor-Bendixson ranks. In the point of view of the authors, the investigation of such spaces will need new tools and ideas.