Dimension-like functions and universality (Q952628)

The authors define relative modifications of the dimension functions dm and DM of \textit{A. K. O'Connor} [Acta Math. Hungar. 55, 83--95 (1990; Zbl 0715.54024)]. The new dimension functions are defined relatively to classes of subsets and classes of bases. Some classes of spaces having dimension dm or DM less or equal to \(\kappa\) need not have a universal space. It is proved that the classes of spaces having a new dimension less or equal to \(\kappa\) are saturated in the sense of [\textit{S. D. Iliadis}, Universal Spaces and Mappings. North-Holland Mathematics Studies 198. (Amsterdam): Elsevier. (2005; Zbl 1072.54001)], which implies that they have universal spaces. That assertion applies e.g. to the classes of completely regular spaces of weight at most \(\tau\) (and, moreover, being (strongly) countably dimensional, or locally finite-dimensional, etc.).