In the paper under review the authors introduce two topological cardinal invariants, defined in terms of sums of spaces belonging to a fixed class. Given a class \(\mathbb{A}\) of topological spaces and an arbitrary space \(X\), the cardinal invariant \(\mathbf{U}_\mathbb{A}(X)\) is defined as being the smallest cardinal number \(\kappa\) such that \(X\) can be represented as a union of \(\kappa\) spaces belonging to the class \(\mathbb{A}\) – notice that if \(X\) cannot be represented as a union of spaces belonging to \(\mathbb{A}\) then \(\mathbf{U}_\mathbb{A}(X) = 0\). Similarly, the cardinal invariant \(\mathbf{U}_\mathbb{A}^{\mathrm{cl}}(X)\) is defined as being the smallest cardinal number \(\kappa\) such that \(X\) can be represented as a union of \(\kappa\) closed subspaces belonging to the class \(\mathbb{A}\) (and, again, if such a representation is impossible then \(\mathbf{U}_\mathbb{A}^{\mathrm{cl}}(X) = 0\)).
Such cardinal invariants are used to define two new classes of topological spaces: for fixed \(\mathbb{A}\) and \(\kappa\) as above, the authors investigate (in the context of universality issues) the classes of spaces given by \(\{X: \mathbf{U}_\mathbb{A}(X) \leqslant \kappa\}\) and \(\{X: \mathbf{U}_\mathbb{A}^{\mathrm{cl}}(X) \leqslant \kappa\}\).
The main results of the paper establish that whenever \(\mathbb{A}\) is a saturated class – in the sense of S. Iliadis [Topology Appl. 107, No. 1–2, 97–116 (2000; Zbl 0986.54021)] – then both the above defined classes are also saturated. A number of related results are also established.