information:classification
Closed unbounded sets and classes

A class C {\displaystyle C} of ordinals is said to be unbounded, or cofinal, when given any ordinal α {\displaystyle \alpha }, there is a β {\displaystyle \beta } in C {\displaystyle C} such that α < β {\displaystyle \alpha <\beta } (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F {\displaystyle F} is continuous in the sense that, for δ {\displaystyle \delta } a limit ordinal, F (δ) {\displaystyle F(\delta)} (the δ {\displaystyle \delta } -th ordinal in the class) is the limit of all F (γ) {\displaystyle F(\gamma)} for γ < δ {\displaystyle \gamma <\delta }; this is also the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent).

Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of ε ⋅ {\displaystyle \varepsilon _{\cdot }} ordinals, or the class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.

A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.

Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal α {\displaystyle \alpha }: A subset of a limit ordinal α {\displaystyle \alpha } is said to be unbounded (or cofinal) under α {\displaystyle \alpha } provided any ordinal less than α {\displaystyle \alpha } is less than some ordinal in the set. More generally, one can call a subset of any ordinal α {\displaystyle \alpha } cofinal in α {\displaystyle \alpha } provided every ordinal less than α {\displaystyle \alpha } is less than or equal to some ordinal in the set. The subset is said to be closed under α {\displaystyle \alpha } provided it is closed for the order topology in α {\displaystyle \alpha }, i.e. a limit of ordinals in the set is either in the set or equal to α {\displaystyle \alpha } itself.