Kerodon

Example 4.7.3.15. Let $(T, \leq )$ be a nonempty linearly ordered set with no largest element. Then the cofinality $\kappa = \mathrm{cf}(T)$ is a regular cardinal. To see this, choose a well-ordered set $(S, \leq )$ of order type $\kappa $ and a cofinal function $f: S \rightarrow T$. Proposition 4.7.2.14 guarantees that $\kappa $ is a cardinal, and Example 4.7.1.31 shows that $\kappa $ is infinite. If it is not regular, then there exists a cofinal map $g: R \rightarrow S$, where $(R, \leq )$ is a well-ordered set of order type $\alpha < \kappa $. This contradicts the definition of $\kappa = \mathrm{cf}(T)$, since the composite map $(f \circ g): R \rightarrow T$ is cofinal.