An important notion in the study of infinite-dimensional manifolds is that of absorbing set. If \({\mathfrak C}\) is a class of compacta, then the subset \(E\) of \(\ell_ 2\) is an absorbing set with respect to \({\mathfrak C}\) if \(E\) is the union of elements of \({\mathfrak C}\) and if for each \(K\in{\mathfrak C}\), closed subset \(B\) of \(K\), and map \(f:K\to\ell_ 2\) such that \(f| B\) is an embedding into \(E\), \(f\) can be approximated by embeddings into \(E\) that agree with \(f\) on \(B\). A result of D. Henderson and A. Pełczyński that appears in Chapter VIII, Section 5 of C. Bessaga and A. Pełczyński’s book [Selected topics in infinite-dimensional topology (1975; Zbl 0304.57001)] shows that there are uncountably many topologically distinct, countable dimensional, \(\sigma\)-compact pre-Hilbert spaces \(E\) in \(\ell_ 2\) that are absorbing sets with respect to the class of all compacta embeddable in \(E\). Here the authors use these spaces to show that for each ordinal \(\alpha\) there is an ordinal \(\beta\geq\alpha\) and a pre-Hilbert space \(E_ \beta\) in \(\ell_ 2\) that is an absorbing set with respect to the class of all compacta of transfinite dimension less than \(\beta\). It is further shown that \(E_ \beta\) is topologically equivalent to \(E_ \beta\times E_ \beta\).