Is $\lambda \in E_{\lambda}^{\kappa}$?

Question

In the chapter on stationary sets in the handbook of set theory, for $\kappa$ a regular uncountable cardinal and $\lambda < \kappa$ regular, Jech defines $$E_{\lambda}^{\kappa} = \{\alpha < \kappa \mid \operatorname{cf}(\alpha)=\lambda \}.$$ Now, it seems that $\lambda$ itself should be a member of $E_{\lambda}^{\kappa}$ as it satisfies $\operatorname{cf}(\lambda)=\lambda<\kappa$. But a paragraph later, it is claimed that "$\bigcup_\lambda E_{\lambda}^{\kappa}$ is the set of all singular limit ordinals less than $\kappa$", so $\lambda$ is not there.

Am I missing something?

Answer 1

You're correct about this. $\lambda$ is in fact of cofinality $\lambda$, so it should be in $E^\kappa_\lambda$. But any other ordinal in $E^\kappa_\lambda$ is singular of cofinality $\lambda$, and removing one element from a stationary set is entirely inconsequential.