Club set

Remarks

Example: the set $C$ of limit ordinals below $\omega_1$ is club in $\omega_1$. It is unbounded, since $\beta + \omega \in C$ for each $\beta < \omega_1$. It is closed, since a limit of limit ordinals is a limit ordinal. Non-example: the set of successor ordinals below $\omega_1$ is unbounded but not closed. Indeed the successor ordinals below $\omega$ have supremum $\omega$, which is not a successor. Condition 1 says that $C$ is closed in the order topology of $\kappa$. If $\kappa$ is a regular uncountable cardinal, the intersection of fewer than $\kappa$ club sets in $\kappa$ is club. For such $\kappa$, the diagonal intersection $\set{\alpha < \kappa : \alpha \in \bigcap_{\xi < \alpha} C_\xi}$ of a sequence $(C_\xi)_{\xi < \kappa}$ of club sets is club. For such $\kappa$, the range of a normal function from $\kappa$ to $\kappa$ is club, and so is its set of fixed points.