In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that 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".
is a limit ordinal, then a set
α < κ ,
α < κ ,
α < β .
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.
is an uncountable initial ordinal, then the set of all limit ordinals
α < κ
In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous).
) is club if every union of a subset of
be a limit ordinal of uncountable cofinality
α < λ
: ξ < α ⟩
be a sequence of closed unbounded subsets of
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.
and for each n < ω choose from each
Since this is a collection of fewer than
their least upper bound must also be less than
This process generates a countable sequence
The limit of this sequence must in fact also be the limit of the sequence
and therefore this limit is an element of the intersection that is above
which shows that the intersection is unbounded.
-complete proper filter on the set
is a regular cardinal then club sets are also closed under diagonal intersection.
closed under diagonal intersection, containing all sets of the form
α < κ
must include all club sets.