In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

 and set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, a club set is a subset of a limit ordinal which is closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 under the order topology

Order topology

In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...

, and is unbounded relative to the limit ordinal. The name club is a contraction of closed and unbounded.

Formal definition

Formally, if is a limit ordinal, then a set is closed in if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

 for every , if , then . Thus, if the limit of some sequence

Limit of a sequence

The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

 in is less than , then the limit is also in .

If is a limit ordinal and then is unbounded in if and only 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

First uncountable ordinal

In mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum of all countable ordinals...

; 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

Normal function

In axiomatic set theory, a function f : Ord → Ord is called normal iff it is continuous and strictly monotonically increasing. This is equivalent to the following two conditions:...

  (i.e. increasing and continuous).

The closed unbounded filter

Let be a limit ordinal of uncountable cofinality

Cofinality

In mathematics, especially in order theory, the cofinality cf of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A....

  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

Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...

 on

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

Diagonal intersection

Diagonal intersection is a term used in mathematics, especially in set theory.If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alphaDiagonal intersection is a term used in mathematics, especially in set theory....

.

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.