Club set

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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[edit]

Formally, if is a limit ordinal, then a set is closed in if and only if for every , if , then . Thus, if the limit of some sequence from is less than , then the limit is also in .

If is a limit ordinal and then is unbounded in if for any , there is some such that .

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 is closed unbounded in ( regular). In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous).

More generally, if is a nonempty set and is a cardinal, then is club if every union of a subset of is in and every subset of of cardinality less than is contained in some element of (see stationary set).

The closed unbounded filter[edit]

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.

See also[edit]