Exercise 11, § 3

A cardinal c is said to be an Ulam cardinal (or to be ulamian) if every gauge on a set with cardinal c is atomic. Show that every countable cardinal is ulamian, that if c is an Ulam cardinal then so is every cardinal $c' \leq c$, and that if $(c_i)_{i \in I}$ is a family of Ulam cardinals such that $\mathrm{Card}(I)$ is ulamian, then the cardinal $\sum_{i \in I} c_i$ is ulamian. Finally, the smallest uncountable cardinal is ulamian (cf. Exer. 10).