Club Set - The Closed Unbounded Filter

The Closed Unbounded Filter

Let be a limit ordinal of uncountable cofinality For some, let be a sequence of closed unbounded subsets of Then 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 and for each n<ω choose from each an element which is possible because each is unbounded. Since this is a collection of fewer than ordinals, all less than their least upper bound must also be less than so we can call it This process generates a countable sequence The limit of this sequence must in fact also be the limit of the sequence and since each is closed and is uncountable, this limit must be in each and therefore this limit is an element of the intersection that is above which shows that the intersection is unbounded. QED.

From this, it can be seen that if is a regular cardinal, then is a non-principal -complete filter on

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

In fact, if is regular and is any filter on closed under diagonal intersection, containing all sets of the form for then must include all club sets.