Club set

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded".

Formal definition

Formally, if \kappa is a limit ordinal, then a set C\subseteq\kappa is closed in \kappa if and only if for every \alpha<\kappa, if \sup(C\cap \alpha)=\alpha\ne0, then \alpha\in C. Thus, if the limit of some sequence from C is less than \kappa, then the limit is also in C.

If \kappa is a limit ordinal and C\subseteq\kappa then C is unbounded in \kappa if for any \alpha<\kappa, there is some \beta\in C such that \alpha<\beta.

If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).

For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded. The set of all limit ordinals \alpha<\kappa is closed unbounded in \kappa (\kappa regular). In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous).

More generally, if X is a nonempty set and \lambda is a cardinal, then C\subseteq[X]^\lambda is club if every union of a subset of C is in C and every subset of X of cardinality less than \lambda is contained in some element of C (see stationary set).

The closed unbounded filter

Let \kappa \, be a limit ordinal of uncountable cofinality \lambda \,. For some \alpha < \lambda \,, let \langle C_\xi : \xi < \alpha\rangle \, be a sequence of closed unbounded subsets of \kappa \,. Then \bigcap_{\xi < \alpha} C_\xi \, is also closed unbounded. To see this, one can note that an intersection of closed sets is always closed, so we just need to show that this intersection is unbounded. So fix any \beta_0 < \kappa \,, and for each n<ω choose from each C_\xi \, an element \beta_{n+1}^\xi > \beta_{n} \,, which is possible because each is unbounded. Since this is a collection of fewer than \lambda \, ordinals, all less than \kappa \,, their least upper bound must also be less than \kappa \,, so we can call it \beta_{n+1} \,. This process generates a countable sequence \beta_0,\beta_1,\beta_2,\dots \,. The limit of this sequence must in fact also be the limit of the sequence \beta_0^\xi,\beta_1^\xi,\beta_2^\xi,\dots \,, and since each C_\xi \, is closed and \lambda \, is uncountable, this limit must be in each C_\xi \,, and therefore this limit is an element of the intersection that is above \beta_0 \,, which shows that the intersection is unbounded. QED.

From this, it can be seen that if \kappa \, is a regular cardinal, then \{S \subset \kappa : \exists C \subset S \text{ such that } C \text{ is closed unbounded in } \kappa\} \, is a non-principal \kappa \,-complete filter on \kappa \,.

If \kappa \, is a regular cardinal then club sets are also closed under diagonal intersection.

In fact, if \kappa \, is regular and \mathcal{F} \, is any filter on \kappa \,, closed under diagonal intersection, containing all sets of the form \{\xi < \kappa : \xi \geq \alpha\} \, for \alpha < \kappa \,, then \mathcal{F} \, must include all club sets.

See also

References

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