In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
and
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, a club set is a subset of a
limit ordinal that is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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
Formally, if
is a limit ordinal, then a set
is ''closed'' in
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
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
In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. W ...
; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.
If
is an uncountable
initial ordinal, then the set of all limit ordinals
is closed unbounded in
In fact a club set is nothing else but the range of a
normal function
In axiomatic set theory, a function ''f'' : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two c ...
(i.e. increasing and continuous).
More generally, if
is a nonempty set and
is a
cardinal
Cardinal or The Cardinal may refer to:
Animals
* Cardinal (bird) or Cardinalidae, a family of North and South American birds
**'' Cardinalis'', genus of cardinal in the family Cardinalidae
**'' Cardinalis cardinalis'', or northern cardinal, t ...
, then
(the set of subsets of
of cardinality
) 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 In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three cl ...
).
The closed unbounded filter
Let
be a limit ordinal of uncountable
cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses t ...
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
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinit ...
, then
is a non-principal
-complete proper
filter on the set
(that is, on the
poset ).
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
*
*
*
*
*
References
*
Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. .
*
Lévy, Azriel (1979) ''Basic Set Theory'', Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover.
*
{{Order theory
Ordinal numbers
Set theory