Computability of a topological poset (Q818359)

In the paper [Topology Appl. 148, 63--82 (2005; Zbl 1062.54039)], the author of the paper under review constructed to a set \(\mathbb{Q}\) the poset \({\mathcal P}(\mathbb{Q})/{\sim}= \{\overline{X}=\{Y\subseteq \mathbb{Q}:\;Y\sim X\}\}\). For this poset the partial order is defined as follows: \(\overline{X}\leq_h\overline{Y}\) if \(X\leq_hY\), where for subspaces \(X,\,Y\subseteq\mathbb{Q}\), \(X\leq_hY\) means that \(X\) is homeomorphic to a subspace of \(Y\) and \(X\sim Y\) means \(X\leq_hY\leq_hX\). In [loc. cit.], it was established that \({\mathcal P}(\mathbb{Q})/{\sim}\) is essentially determined by considering only the scattered subset \(X\subseteq \mathbb{Q}\) of finite Cantor-Bendixson rank \(N(X)\). For the purpose of the present paper the author considers only scattered countable metric spaces with finite \(N(X)\) and shows that the set \(\mathbb{A}=\{X\subseteq \mathbb{Q}:\;1<N(X)<\omega\}/{\sim}\) can be generated by a computer program. As an example, a complete list of homeomorphism types of spaces of rank \(\leq 4\) and a presentation of the embeddability ordering of these spaces is given.