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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".

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Formal definitionEdit

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 filterEdit

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 alsoEdit

ReferencesEdit