closed unbounded set,regular cardinals,cofinality

Question

Given two regular cardinals $\lambda>\mu$, why this set is a closed unbounded set in $\lambda$?

{$\alpha$ | cf($\alpha$)=$\mu$ , $\alpha<\lambda$}

Answer 1

It isn’t necessarily. It’s always unbounded, but it need not be closed. For example, take $\mu=\omega$ and $\lambda=\omega_2$: $\omega_1$ is a limit point of $\{\alpha<\omega_2:\operatorname{cf}\alpha=\omega\}$, but it’s not in that set. Or take $\lambda=\omega_3$ and $\mu=\omega_1$; if $A=\{\alpha<\omega_3:\operatorname{cf}\alpha=\omega_1\}$, both $\omega_1\cdot\omega$ (ordinal product) and $\omega_2$ are limit points of $A$ that are not in $A$.